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Essential Algebra & Trig Review Before Calculus 1 | Everything You Need to Know — Transcript

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0:00Welcome to Math with Professor V. If
0:02you're getting ready to take calculus 1,
0:05or you've already started and realized
0:07your algebra and trig skills could use a
0:09little review, this video is for you.
0:12We're going to go through the essential
0:13pre-calculus topics that I want my
0:15students to be comfortable with before
0:17diving into calculus. We're going to
0:18review functions, function notation,
0:21domains, transformations, composition
0:23and inverse functions, exponential and
0:26logarithmic functions, trigonometry,
0:30inverse trig functions, and plenty of
0:32algebra along the way. So, this is a
0:34comprehensive review. It's a little bit
0:36of a longer video. You can work through
0:38the entire thing with me or use the
0:41timestamps in the description to jump to
0:43the topics that you need the most help
0:44with. And if you want to follow along
0:46using the lecture template I'm working
0:48through in this video, I've included it
0:50as a downloadable resource in my
0:52complete calculus 1 course at
0:54professorvmath.com.
0:56So go grab some paper, get a snack
0:58maybe, and work the examples along with
1:01me to make sure your algebra and trigs
1:03are ready for calculus 1. So much of
1:06calculus is the study of functions.
1:09That's why we want to make sure we have
1:11a solid understanding of functions
1:12before we begin. And functions can be
1:15thought of and represented in a variety
1:17of ways. What's the definition? First
1:20things first, a function f is a
1:23collection of ordered pairs in which
1:26each element x is in a set d. D is the
1:30domain of f and it's assigned to exactly
1:34one element called f ofx from a set e
1:38called the range. So, of course,
1:41functions are also commonly represented
1:43using graphs. In this case, the vertical
1:46line test can be used to determine if
1:48the graph represents a function. So,
1:50remember, if you are given the graph of
1:52a function, a vertical line should never
1:54cross the graph at more than one place
1:57at a time. Okay. So, the graph of a
2:00function f ofx is shown. They're already
2:02telling me it's a function. You can
2:03verify. Just kind of imagine running a
2:06vertical line through the graph. Yeah,
2:07it's never hitting anywhere more than
2:09once. State the value of f of one. So
2:12this is where it's really important that
2:13you understand function notation.
2:15Remember the function is represented by
2:16f ofx. So whatever's inside those
2:19parentheses that represents the x value.
2:23And then f ofx basically you could think
2:25of it as y y y y y y y y y y y y y y y y
2:27y y y y y y y y y y y y y y y y y y y y
2:27y y y y y value. So they're saying
2:28what's the y value when x is one? So you
2:31come over to the graph and you go okay x
2:33is one right here. So let me come find
2:36what the y value is on the function on
2:38the graph when x is one. Can you tell
2:40what it is?
2:42It's three. Exactly. So f of 1 is equal
2:46to 3. And the reason why this is more
2:49useful to us than just saying y is
2:51three, right? Because I just told you
2:53right now f ofx and y basically mean the
2:56same thing. It's because if I tell you f
2:59of 1 equals 3, not only do you know,
3:02okay, y is 3, but you know y is three
3:04when x is 1. If I just came up to you
3:06and said, hey, y is three. You'd be
3:08like, okay, that's fantastic, but I
3:10don't know where that happens on the
3:11graph. This has so much more
3:12information. That's one plus amongst
3:14many more for using function notation.
3:17Okay, next part. For what values of x is
3:22f ofx equal to 1? Remember your brain,
3:24you look at this, you go, oh, y is equal
3:26to 1. Now they want to know when does
3:28that happen for what values of x? So y
3:30is equal to 1. Right here I'm going to
3:33just draw the line through where y is 1.
3:36What is x? That happens here and here on
3:40the graph. So it looks like when x is -2
3:44and when x is 2 and 1/2, right? 2.5.
3:49Okay. Lastly, state the domain and range
3:52of f. Now notice there's not arrows on
3:55the end points of the graph. So what we
3:56see is all there is. That's it. That's
3:59all she wrote folks. Here is the graph
4:02of f and in its entirety. Okay. So the
4:05domain is going to be the set of all of
4:07the x values that the graph attains. So
4:10let's see. This is x=1
4:142 3 4 -4. And yep it obtains every
4:18single x value up until where is this?
4:21x= 1 2 3. So all the way from -4 to 3.
4:26-4 to 3. This is interval notation.
4:28You're going to use it more extensively
4:30in calculus than set builder. But if
4:33you're used to set builder, you would
4:35write your set, declare your variable x
4:37such that -4 is less than or equal to x
4:40is less than or equal to 3. Okay, set
4:43builder has its place, but you'll notice
4:45in calculus we more heavily use interval
4:48notation. Now, the range, we're looking
4:51at the y values that the graph takes on
4:53or achieves. Okay, so this is y= -1. Any
4:57gaps? Nope. Everything's good, good,
4:59good, good, good. Up until where's this?
5:01Y equ= 1 2 3 4 5. So from -1 to 5. And
5:07notice I'm putting brackets because
5:08these end points are closed circles.
5:11Mhm. So they're included.
5:14If you were to write this in set
5:15builder, range is all the y values such
5:19that1 is less than or equal to y is less
5:21than or equal to 5. Like that. Okay,
5:24good. Another thing you're going to
5:26study extensively is where your function
5:28is increasing and where it's decreasing.
5:30So, let's make sure we understand these
5:32ideas very thoroughly. A function is
5:35increasing on an interval I if f ofx1 is
5:39less than f ofx2 when x1 is less than x2
5:43on the interval. Don't freak out when
5:45you read a math definition, okay? It's
5:47telling you a lot of info. You just got
5:48to break it down piece by piece. Okay?
5:51They're saying f ofx1 that means the y
5:54value when x is 1 is less than f ofx2
5:58the y value at x2 whenever x1 is less
6:02than x2. I'm going to draw it for you
6:05cuz a picture is worth a thousand words.
6:07So I can just talk less show you and
6:09you'll get it. Draw it. So the y value
6:12at x1 is less than the y value at x2.
6:16y value at x1 less than the y value at
6:20x2. Didn't I fulfill exactly what they
6:24said? Uh-huh. And this is the definition
6:26for increasing. Oh yes. See how my
6:29function is increasing from x1 to x2.
6:33Exactly. And then a function is
6:36decreasing. So let me let me color code
6:38for you. Look at this. Increasing. Then
6:40we say a function is decreasing if f
6:43ofx1 is greater than f ofx2. So the y
6:46value at x1 needs to be bigger than the
6:49y value at x2.
6:51That's all I need to know. And then the
6:53function's decreasing. Yep, that checks
6:56out. Okay, great. So for the graph in
7:00the example above, the one that we just
7:02did a second ago, on what interval is f
7:05increasing? So when they ask you for
7:08intervals of increase and decrease, you
7:10always give values in terms of x. in
7:14terms of x.
7:18The interval is in terms of x.
7:20Increasing already describes what the
7:22function what y is doing. So you have to
7:25tell me where is it in this case
7:27increasing. So where in terms of x is
7:30the function going up? And always read
7:32your graph left to right. So imagine you
7:35can imagine you're walking on the graph.
7:36Why not? Here we go. Dah. Oh, we're
7:40going downhill. That means we're
7:42decreasing right here. Things change for
7:44us though. Yes. Now the treadmill is on
7:47an upward incline. We're going uphill.
7:49So we're increasing on this interval.
7:52Where is it? Give the x values. Where is
7:56this? Where is this? This is at x= -2.
7:59This is at x= 1. Okay. And after that
8:03it's just decreasing, decreasing,
8:04decreasing the rest of the way. So from
8:07-2 to 1. I'm going to put these as open
8:11intervals. That's the way I was taught.
8:14That follows with the textbook that we
8:16use. Believe it or not, this can be up
8:18for debate among some math professors or
8:23mathematicians, but typically
8:26right where the graph switches from
8:28decreasing to increasing is a turning
8:31point. It's doing neither exactly at
8:33that point. So, it's not included on
8:35intervals of increase or decrease. Okay.
8:39Good. So, make sure you give x values
8:40for that one and you're going to be in
8:42business. That's a common mistake I see
8:44my students making all the time. So,
8:46that's why I'm warning you now. All
8:48right. It's also extremely important
8:50that you're able to use, understand, and
8:54correctly manipulate function notation.
8:57So, here I'm giving you a function. Let
8:59f(x)=
9:012x^2 + 5x - 3. Let's find the following.
9:05F of 2. So that's asking me to
9:07substitute in 2 everywhere I see x in my
9:10function. So that would equal 2 * 2^ 2 +
9:155 * 2 - 3. Not bad, right? So 2 * 2^ 2.
9:22What's that going to be? That's going to
9:23be 8 + 10 - 3. 18 - 3. That's 15. Okay.
9:30So far I think everyone's feeling fine,
9:32right? F of a + h. now gets a little
9:35spicier, but don't freak out. Last time
9:37we had two inside here, so I substituted
9:40two everywhere I saw x. Okay, now
9:42they're just saying instead of two, we
9:44need to substitute in a + h. So just
9:46keep it together. You can do this. So we
9:48have 2 * the quantity a + h 2 + 5 * yep,
9:55a + hus
9:583. That's it. Look at us go. Okay,
10:00follow your order of operations. So I'm
10:02going to multiply this all out.
10:03exponents first before I distribute that
10:062 through. So this is 2 * a^2 + 2 ah + h
10:122
10:14plus now we can distribute this 5 a + 5
10:17h - 3 and then now let me distribute the
10:20two through. So 2 a 2 + 4 ah + 2 h 2 + 5
10:28a + 5 h - 3.
10:32All right, good job.
10:35Now for the creme de la creme. f of a +
10:38h minus f of a over h. No, this is not
10:41some random creepy looking expression.
10:44This is the difference quotient. You
10:47should have seen it in pre-calc.
10:50They might have told you a little bit
10:51about what it represents or what it
10:53does. You'll study that more in calculus
10:56in the following units. Okay, let's just
10:58make sure we can actually evaluate it
11:00and simplify it. Let's make sure our
11:03algebra is on point. Okay, f of a plus h
11:06minus f of a. The good thing is they
11:08kind of held our hand into this one. We
11:10already computed f of a plus h right
11:12here in part b. So I can just copy down
11:16that result. Let me switch color so we
11:18all know what's going on. We're doing a
11:19new part of the problem. F of a + h is
11:22right here. 2 a 2 + 4 ah h + 2 h 2 + 5 a
11:30+ 5 h - 3
11:34and then I have to subtract
11:38f of a. Well f of a don't freak out.
11:41You're just going to replace all the x's
11:42with a's here. No big deal. We got this.
11:45That would be 2 a^ 2 + 5 a - 3 and then
11:51over h.
11:53Okay. How do you know you're done
11:54simplifying when working with the
11:56difference quotient? This h down here
11:58needs to cancel.
12:01Sure does. And it always will.
12:04You should always be able to. Okay.
12:07Notice here there's just a massive
12:09cancellation party happening in the
12:10numerator. There sure is. 2 a^ 2 is
12:13going to cancel with this - 2 a^2 see
12:15how this is going to distribute
12:185 a cancels 3 cancels in fact I'm going
12:22to give you a hint that you can use
12:25whenever you simplify
12:27difference quotient anybody without an h
12:31should cancel out so these all have h's
12:35in them so I'm like oh they're probably
12:36going to stay okay so what do we have
12:38left
12:404 ah h + 2 h^2 + 5 h over h.
12:48And we're almost there. I told you the
12:50goal is to cancel out that h in the
12:51bottom. You could cancel term by term by
12:56term if you're pretty comfortable. I
12:58think it's a little dangerous sometimes.
13:00So, I would recommend to my students
13:02that they factor out the h first
13:06and then cancel. Okay? Just so you're
13:08not tempted to do something illegal. I
13:09don't want the math police to come
13:11snatch you. Okay. And then this is just
13:14going to be 4 a + 2 h + 5. Good. Okay.
13:19What it represents, what we do with it,
13:22you can wait until you start your first
13:25unit of calculus.
13:27Now, moving on. In most cases, the
13:30domain of a function is the set of
13:33values of x for which the function makes
13:35sense and yields a real number itself
13:37for which it's defined. So let's look at
13:40some examples like if you have a
13:42polomial,
13:44we've been dealing a lot with polomials
13:45up until now. x^2 + 3x - 7.
13:49Are there any is there any reason why
13:52you couldn't substitute in a particular
13:53value of x and get a value for y that
13:56makes sense that's well defined? No
13:58polomials are beautiful and easy to work
14:00with in that way. So in this case the
14:02domain is all real numbers. You could
14:04also write it in interval notation
14:05negative infinity to infinity. Now
14:08another kind of function is a rational
14:10function.
14:12Let me give you an example of one. It's
14:14when you have a polomial divided by
14:15another polomial. Now is there any
14:17reason we couldn't have a value in the
14:20domain? Is there a value of x that we
14:22wouldn't be allowed to plug in because
14:25it wouldn't make sense? we wouldn't get
14:26a y-value that's well defined.
14:29Yes, I can't have a zero in the
14:31denominator. Dividing by 0 is undefined.
14:33So in this case, my domain is all x's
14:36such that x doesn't equal -3.
14:39Okay, so basically for a rational
14:41function, you have to exclude any values
14:43that would make the denominator zero.
14:45You could also write that in interval
14:46notation. You would say infinity to -3
14:49with parentheses
14:51union -3 to infinity. That's how you
14:53jump over it basically.
14:56Okay, another kind of function that'll
14:57come up a lot is a radical function. And
15:01if you have a radical function with an
15:03even index, meaning this number here is
15:06even. So 2 4 6 8. When it's a two, we
15:09don't write it though, right? It's
15:11assumed to be two.
15:13Well, we can't take square roots of
15:14negative numbers. So we require
15:16whatever's underneath the radical,
15:18that's called the radicand to be
15:20positive or it could be zero, right?
15:22Square root of 0 is zero. So if you
15:24solve this inequality, you get that x
15:26has to be less than or equal to 3. So
15:28that's our domain. We do have a
15:30restriction here. So you could say this
15:32is set builder notation. x such that x
15:35is less than or equal to 3. Or if you
15:38want to write it in interval notation,
15:39that would beative infinity to 3 and I
15:42can include three square root of 0 0. So
15:45don't worry, as long as it's not in the
15:46denominator, you're good to go.
15:49Okay,
15:51let's shift our gears now and talk about
15:52peace-wise defined functions. They're
15:55also very useful. They occur commonly in
15:58the real world. Perhaps the one with
16:01which you're most familiar is absolute
16:04value of x. And maybe you haven't seen
16:06absolute value of x written as a
16:07peace-wise defined function, but this is
16:09how you're going to work with it
16:11extensively in calculus. So you probably
16:14think, oh, absolute value tells you the
16:17distance that a number is from zero on
16:19the number line or it takes negative
16:21values and makes them positive, right?
16:23But we can also define it peacewise. We
16:26can say
16:27f ofx is equal to x if x is greater than
16:31or equal to zero and it's equal to the
16:33opposite of x if x is less than zero.
16:37Let's stop and think about this for a
16:38second. Basically, this peace-wise
16:40function is telling me if x is already
16:44greater than or equal to zero, so it's
16:46positive or zero. When you plug it into
16:48absolute value, don't do anything. It
16:51stays the same.
16:53If x is negative, if x is less than zero
16:56and you're finding its absolute value,
16:59multiply by a negative. And what happens
17:01if we take a number that's negative and
17:02multiply by another negative? It's going
17:05to make it positive. Exactly. So from
17:08here on out, the majority of the time in
17:10your calculus journey, when you see
17:12absolute value of some expression and
17:14you're asked to find its limit,
17:16derivative, or integrate it, you want to
17:19rewrite it as a peace-wise function.
17:21This is an essential skill and it's
17:23often missed in the curriculum. So I
17:25have a whole video dedicated to this
17:27topic. I'm going to link it in the
17:28description. It's that important that
17:30you know how to do it. But basically,
17:32say in a month or so, or maybe not even
17:35that long, you're given some function g
17:37ofx= absolute value of x - 3 and you're
17:40asked to find limits or derivatives.
17:42This is stuff you're going to do in
17:43calculus. You want to be able to rewrite
17:46it as follows as x - 3 if x is greater
17:50than or equal to 3 or the opposite of x
17:52- 3 if x is less than 3. So, you have to
17:56figure out where to break up the domain.
17:57That happens wherever the expression
17:59inside is zero. And then you have to
18:01figure out which um portion of the
18:03peacewise function is equal to the
18:05original and which one you need to
18:06multiply by a negative. And I'll walk
18:08you through all of that in the video.
18:09But make sure you know how to do this.
18:11In fact, let me just put a double star.
18:14Find some time to get that skill under
18:16control. Okay, good. If I talked about
18:20everything that you needed here, this
18:21video would be like 4 hours. Okay, let's
18:24find the domain of the following
18:25peacewise function and sketch the graph.
18:28So they're telling me f ofx is equal to
18:30x + 3 if x is less than -1. And notice
18:34this inequality is not inclusive. So I'm
18:36going to draw the line x + 3 all the way
18:39up until -1 and then open circle right
18:43there at 1. You know what I mean? And
18:46then x^2 that's a parabola. If x is
18:48greater than1 again it's not including
18:51it. So we're just going to have a open
18:52circle there. So it looks like all real
18:55numbers are included except for poor
18:57little negative one. So h okay. So the
19:01domain is all x's such that x is not
19:04equal to negative1. I don't know what
19:06negative1 did to anybody but that's
19:07fine. Let's go ahead and graph. So
19:12graphing peacewise functions is really
19:13hard for students sometimes. You can you
19:16don't have to make a table. You guys
19:17don't. But I'm just going to show you if
19:18you're kind of struggling.
19:20Um, wherever the domain splits, I list
19:23that value twice on the table. Okay,
19:26negative one twice. And then do a couple
19:28smaller and a couple bigger. Since just
19:31a line, I feel good about it. Okay,
19:33these I'm going to plug into
19:36let's see. So -2,1 those are values that
19:40are less right than negative 1. So I'm
19:42going to plug them into x + 3. And make
19:45sure when you plot it at negative 1, you
19:47put an open circle because this is not
19:49included. And then the other values I'm
19:52going to plug them into because they're
19:53bigger than negative 1, right? 0 1. So
19:56they go in this part of the peiewise
19:57function. So I'm going to plug them into
19:59x squar. But make sure to open circle.
20:01So if you want to make a table, you can.
20:03You don't have to.
20:05Let's see. [sighs]
20:07-2 + 3 that's 1.
20:11-1 + 3 that's 2.
20:161^2 1 0 1. Okay, let's plot this bad
20:20boy. So, we've got -2 1
20:25-12 open circle and it's a line just
20:29forever to the left, right, for all the
20:31values smaller than 1. So, just draw a
20:33line. And this time I will put an arrow
20:36to indicate it keeps going, going,
20:37going. It should have gone throughg3.
20:40Hold on, let me straighten that up.
20:42There we go.
20:44So, come on.
20:46Yeah, I knew that the x intercept was
20:48going to be -3 just by looking at the
20:50function. Okay, then this next chunk is
20:52the parabola -11 open circle
20:56then 0 0 1 1 and then it continues on
20:59the other side.
21:01Cute. Okay, perfect.
21:05Let's see what's the range. I know they
21:08didn't ask for it, but we're
21:09overachievers. Okay, what y values are
21:12attained? Well, this goes down forever.
21:15Okay. Okay. Now, there's this little
21:17hole here, but that's fine because one
21:20is achieved on this portion of the
21:21graph. Up, up, up, up. There's a hole
21:23here, but it's fine. It picks back up.
21:25So, all real numbers. Oh, yeah. That's
21:28the range. All real numbers.
21:31Okay. I have another video dedicated to
21:33helping you figure out how to graph
21:34peace-wise functions if you if this is
21:37something that you know you struggle
21:38with. And you know the goal is
21:40eventually to not need this table but if
21:42you kind of don't use it as a crutch but
21:44if it helps you just kind of get the
21:46ball going that's great. Okay. Now the
21:49symmetry of a function is also
21:51important. We say that a function is an
21:54even function. If f ofx is equal to f
21:59ofx for all x in its domain. And if a
22:03function is even then that means it is
22:06symmetric
22:08across the y- axis.
22:11So you oh what am I writing? Okay across
22:15the y ais you know lots of even
22:18functions off the top of your head.
22:20f(x)= x^2
22:23is probably the most famous one. Let's
22:26check it with the definition. f ofx
22:29would be -x^2
22:32which comes out to positivex^2 which is
22:35equal to f ofx exactly. And you could
22:39imagine right your parabola y = x^2 it
22:43does have symmetry about the y ais
22:47but basically the definition tells me
22:49that if we have x y
22:52it should have the same value atx right
22:55same y-value
22:57okay a function is an odd function if f
23:01ofx equals
23:04f ofx you see how the negative came
23:05outside now so the y-value would be the
23:07opposite it sign for all x in its
23:09domain. And if you have an odd function,
23:12it is symmetric
23:15across guess,
23:18not the x-axis. Functions can't be
23:20symmetric across the x-axis. Then they
23:22would violate the vertical line test.
23:24[clears throat] I know it's symmetric
23:26across the origin.
23:28The origin. Yes. Okay. Let's think of
23:32like a very easy odd function that you
23:34probably know, right? x cubed. FX = x
23:41cubed. So f ofx would be -x cubed. Since
23:47it's raised to an odd power and it's an
23:50odd function, go figure. Then this is
23:52going to be -x cubed. And I think you
23:55guys know what it looks like, right?
23:56This guy. And then we have this guy for
23:59x squ. So if it's odd, then you have xy.
24:04And then you also have -x - y. So it's
24:10reflected over here and here. That's why
24:13it's origin symmetry.
24:15Okay. So let's check algebraically. So
24:18without the graph whether each function
24:20is even, odd or neither. So f ofx= x
24:24cubed over x^2 + 1. So just start off
24:28substituting inx for x.
24:32So we have x cubed over
24:36oops do what you're telling them
24:38professor vxx^2 + 1. So in the numerator
24:42I'm still going to have the negative
24:43there cuz three is an odd power but in
24:45the denominator it gets absorbed.
24:48And then let's see is this equal to my
24:50original function? No. Is it the
24:52opposite of my original function? Yes.
24:54Cuz I can take that negative outside and
24:56write this as x cubed over x^2 + 1. So
25:01then this is negative f ofx isn't it?
25:04And by definition
25:06that makes this function odd.
25:09Okay. Now what about h ofx= cosine x?
25:14This one you just have to know. I know
25:16that cosine's an even function. Yes. So
25:19h ofx
25:22is cossine ofx
25:25which equals cossine x which is h of x.
25:29So it's even. And you might say, "Oh,
25:31Professor V, I forgot that." It's okay.
25:33So if all the trig functions cosine of x
25:37and seeant x are even
25:40and the rest are odd. Okay, there. Now
25:43you got it. And it should make sense
25:45seeant is cuz seeant is reciprocal of
25:47cosine, you know. Okay, fantastic.
25:51Now there's a number of types of
25:52functions with which you should already
25:54be somewhat familiar. You should make a
25:57point of being familiar with basic
25:59characteristics of these functions
26:01because they're going to be the building
26:02blocks for more interesting complicated
26:04functions that we're going to explore in
26:06this course. And I'm briefly going to
26:08mention the various types here, but if
26:10you need a more detailed review, then
26:11you can go to the pre-calculus video
26:13lectures playlist on my YouTube channel,
26:15Math with Professor V, or go to my
26:18website, professorvmath.com.
26:21So, first let's start off with linear
26:22functions. Normally you see it as y = mx
26:25plus b, but we're talking about it as a
26:27function. So we're just replacing y with
26:29f ofx. And m is the slope. 0 comma b is
26:34the y intercept. Hopefully that's not
26:36shocking to you. Another kind of
26:38function that we work with extensively
26:40is a polomial function. So polomials
26:43have the following form. These are all
26:46the coefficients. A n a n minus one all
26:49the way down to a z. That would be the
26:51constant term. And then you have x to
26:52the n n n n n n n n n n n n n n n n n n
26:53n n n n n n n n n n n n n n n n n n n n
26:53n n n minus one. You could have some
26:54missing, right? It's called degree n
26:56because n is the highest power on x and
26:59n has to be a non negative integer. So
27:02no fractional exponents, no negative
27:05exponents and a subi are the
27:07coefficients.
27:09Power functions, we just looked at a
27:11couple f ofx= x to the a where a is
27:15constant. If a is even, the graph
27:18resembles x squared.
27:21If a is odd, the graph resembles x
27:25cubed. And then if you put a coefficient
27:27in the front, that could change things
27:29around.
27:30Okay. Rational functions have the
27:32following form. f ofx= p ofx over q ofx
27:37where p and q are polomials. So I always
27:40think of it as rational has the word
27:42ratio in it. This is a ratio of two
27:45polomials. Good.
27:48Now, algebraic functions. We say that a
27:51function is algebraic if it can be
27:54constructed from poloms using algebraic
27:57operations like addition, subtraction,
28:00multiplication, division, taking roots,
28:03etc.
28:04trigonometric functions. You need to be
28:07very familiar with the graphs of s
28:10cosine tangent as well as their
28:12reciprocals cosecant.
28:15And trigonometric functions are
28:17periodic. So know the period of each,
28:18how often they repeat, what the
28:20amplitude is for particular ones, not
28:23all of them have amplitude, etc., etc.
28:26One thing to keep in mind when we're in
28:27calculus, we're going to work with
28:29radians only when um doing trig. Okay?
28:33So get used to that. Exponential
28:36functions have the form f ofx= a to the
28:39x where the base a is a positive
28:42constant.
28:44And just know the general shape. So if a
28:47is greater than one, your exponential
28:50functions graph
28:53would look like this. You would have
28:55exponential growth. All exponential
28:57functions go through the point 0 comma
29:001. That's like home base right there.
29:02And then horizontal asmtote y equals
29:05that one looks disgusting. Hold on.
29:08Get it together. Oh, much better. Okay.
29:12Horizontal asmtote y equals z and it
29:15increases upward. Domain is all real
29:18numbers
29:20and then the range obviously you can see
29:22it's 0 to infinity. Now if a is less
29:25than one then we have exponential decay.
29:28So it still goes through 01. It's just
29:30the graph gets reflected over the y
29:34axis. It sure does right here and then
29:38it goes like this. Mhm.
29:41Okay. Perfect. Logarithmic functions. F
29:45ofx equals log base a of x where the
29:49base a is a positive constant. These are
29:52the inverse functions of exponential
29:54functions. I can't tell you how
29:55important it is for you to know this
29:57basic graph in your head. So, I'm going
29:59to draw it for you right now. Please
30:01burn it in your brain. If you can't, go
30:04get a tattoo on the inside of your wrist
30:06or somewhere. Okay? Yeah, you got to be
30:08that committed to this graph. Okay? So,
30:10since it's the inverse of your
30:12exponential function, remember inverse
30:15functions, you can just interchange any
30:18ordered pair and this will be on the
30:20graph of the inverse. So, instead of 01,
30:23logarithmic functions go through the
30:25point 1 0. And instead of a horizontal
30:29asmtote at y equals 0, they have a
30:31vertical asmtote at x= 0. Mhm. So it
30:35comes down this way and then it
30:38increases not as quickly as an
30:39exponential function but something like
30:41that. So please remember the domain
30:43would be what?
30:460 to infinity. Yes. And the range
30:50all real numbers.
30:52Notice the domain and range are also
30:54interchanged. That's always the case
30:55with inverse functions. This one's so
30:58important. You're going to be doing
30:59problems with limits and other things
31:01and you're going to need to be able to
31:02recall this basic shape in your mind.
31:04Remember that there's this vertical
31:06asmtote here and that the grow graph is
31:09approaching
31:11infinity as you approach zero from the
31:13right. In fact, I'm just I can't resist.
31:16If I were to ask you what is the limit
31:19as x approaches zero from the right
31:22of this function. This is something
31:24that's coming up and here's f ofx.
31:27Basically it's saying if you're walking
31:29along the graph and x um you're heading
31:32towards x equals z from the right side
31:34meaning from this way. So you're walking
31:39where are you going? You're going to
31:41negative infinity. This is super
31:43important. Okay. If this didn't make
31:45sense it's okay. It's coming up, but I
31:46just wanted to give you a sneak preview
31:47and motivate why you need to know this
31:49graph. It's so essential.
31:52Okay. Another reason why we want to have
31:54our catalog of functions ready is so
31:57that we can modify those functions by
31:59translating, reflecting, shrinking,
32:02stretching, or combining the functions
32:04through your various operations. And all
32:07of these skills are going to be
32:08important throughout your mathematical
32:09career. You should have worked with some
32:11transformations previously and be able
32:14to do it without too much stress. So
32:16let's remind ourselves what sort of
32:18different operations are available to
32:20us. So we have vertical and horizontal
32:23shifts or translations depending on your
32:25book. So suppose C is some positive
32:28constant. If you see that your function
32:30has C added to it outside then that
32:34shifts the graph up C units. If C is
32:37being subtracted it shifts the graph
32:38down. If C is being added inside
32:43with X, yeah, then it's affecting the X
32:46values and it actually goes opposite of
32:48what it looks like. So it shifts it left
32:49C units if it's plus C. If you have a
32:52minus C, it shifts the graph right C
32:54units. And then we can also stretch. So
32:57stretching involves multiplication.
32:59Stretch, shrink, and reflect. So if you
33:02multiply by C, that's going to stretch
33:04the graph vertically. If you multiply by
33:071 over C, it shrinks the graph
33:09vertically by a factor of C. If you
33:11multiply by X, it shrinks it
33:14horizontally by a factor of C. Just
33:16think X always looks like the opposite.
33:18Dividing by C stretches it horizontally.
33:21A minus sign outside, think of it like
33:24it's changing all the y values. They're
33:26signs. So if you switch the y-value
33:29sign, it's going to reflect the graph
33:31over the x axis. And then this is
33:34changing all the signs of x. That's
33:35going to reflect the graph about the y
33:37ais. Okay. If you need a review of
33:41transformations, I got you. It's I do a
33:43bunch of it in pre-calc. Okay. The graph
33:47of f is shown. Sketch each of the
33:50following. So we don't have an equation.
33:52We're going to have to figure out y =
33:54fxus 2, f ofx + 2, 12 fx, and f ofx + 1.
34:01So what I'm going to do, let's first
34:02label what these points are. That way we
34:06can just apply the transformations to
34:08those four points and kind of figure out
34:09where everything is a little more
34:10easily. So that's -21.
34:15This is 1 comma 2.
34:18This is 2 comma 2 and then this is 3
34:22comma 1. Right? Okay. So looking at the
34:26transformations y = fxus 2. This means
34:29shift the graph down two units.
34:35So all the y values you're going to
34:37subtract two from them. So now you're
34:40going to have -2
34:43-3. So that would be down here. 1 2 is
34:47going to become 1 0.
34:502 is going to become 2 0.
34:54And then 3 1 is going to become 31.
34:58Okay, not bad, right?
35:03Not bad. Okay, now we have y = f ofx +
35:072. This means we need to shift the graph
35:11left 2 units.
35:14So, we're going to subtract 2 from x
35:16this time. So, -21 is going to become
35:20-4, -1.
35:231 comma 2 subtract two from x it'll be
35:26-12
35:292 is going to become 02
35:33and 3 1 is going to become 1 one
35:38look at that look at us go you don't
35:40have to actually figure out what the
35:41points are if you just feel comfortable
35:43kind of like moving them you know
35:46live your life okay y equals 12 f ofx so
35:51this is A very good shrinking of the
35:55graph of f ofx vertically by a factor of
35:58two. So shrink vertical shrink
36:06I know it's vertical not horizontal
36:08because it's outside so it's affecting
36:09the y value. It's touching f ofx by two.
36:13So you're going to literally just divide
36:14all the y values by two or multiply them
36:16by a half whatever you want to think of
36:18it as. So we'll have -2 - one2
36:22one 2 becomes 1 1 2 2 becomes 2 1 and
36:27then 3 1 becomes 3 1/2. I think it's
36:31helpful to actually transform the
36:32ordered pairs in this case. Okay.
36:35-2gative a2.
36:37Where'd you go there?
36:401 1 2 1 3 1/2 H.
36:46There it is guys. Okay. We shrank it.
36:50Cute. Last one. y = f ofx + one. So
36:54there's multiple transformations that
36:55are happening. Notice this minus sign
36:58here. Okay. It's affecting the x value.
37:01So we're going to reflect across the y
37:03ais.
37:07And then this plus one, it's outside.
37:09It's not in parentheses with the x. See
37:12how it's different than this? This would
37:13be left or right. This is up. Up. Shift
37:16up one unit.
37:19shift. Don't forget the F. Okay. Shift
37:21up one. Now, it matters the order that
37:24you do these in. Oh, yeah. Just think of
37:27your order of operations. Whenever we're
37:29simplifying expressions, we have to
37:31multiply before we add or subtract. So,
37:34looking back here, right, stretching,
37:37shrinking, reflecting, these all involve
37:39multiplication. You have to do this
37:40first before you do any vertical or
37:42horizontal shifting. That involves
37:44addition, subtraction. That's all.
37:45That's how I keep it straight. do
37:47multiplication before addition and
37:49subtraction. Oh, no problem. Then I have
37:51to do this before I shift up. That's
37:54all. So reflect across y. Literally,
37:57you're going to multiply all the x
37:58values by -1.
38:02So what were they? Okay, multiply the x
38:05values by 1. It's going to become 21 -1
38:09-12
38:12-2 2 and -3 1. Should we draw this
38:15intermediary graph? Why not? Or 21
38:20-1 2 -2 2 -3 1. So this is just doing
38:26that first transformation.
38:29Got it? Let me color code for you. This
38:33happened and we got this. Then we still
38:36have to go up one unit. I'm not done. So
38:39now I'm going to add one to the y
38:42values. So it'll be 2 0 -1 3 -2 3
38:50-3 2 or you can literally just come here
38:52and go boop boop boop boop. I'm going up
38:56one unit. That's not hard to do.
38:59Okay, there we go. That's the final
39:01answer. This blue guy. Okay,
39:05fantastic.
39:06So, let's put it into action because
39:09you're not going to be graphing these
39:10like weird random things throughout
39:12calculus. Graph the function by hand,
39:15not by plotting points. Meaning, don't
39:17make a table. Nobody wants your table.
39:19But start with the graph of one of the
39:22standard functions and then apply the
39:24appropriate transformations.
39:26So, here's how I teach it to my
39:28students. This is a topic that we cover
39:29in, you know, intermediate algebra,
39:31pre-calculus, stuff like that. Identify
39:34the parent function without all the
39:36transformations and bells and whistles
39:38going on. So the parent function is y=
39:41roo<unk> of x. Hopefully you have
39:45memorized right what the graph of this
39:47guy looks like. What the parent function
39:49looks like. If not that's okay. You can
39:51go watch my video where I go over like
39:52the main functions. But it goes through
39:550 0 1 1 42. It's the upper half of a
40:00sideways parabola. Okay, so I tell my
40:05students just remember these key points
40:07from the original graph. 0 0 1 1 42. We
40:12just picked those from plugging in
40:14values of x that give us nice values of
40:16y, right? Like I don't want to deal with
40:18two square<unk> of 7. Heck no. Okay.
40:22Now, we're going to go through and
40:23identify what sort of transformations
40:25occurred to my parent function so that I
40:27ended up with this f ofx here. I see
40:30three different transformations.
40:33this one, this -2 that's being
40:36multiplied,
40:38and then this three that's being added.
40:40Yes. So, let's think what order do we
40:43need to do these transformations in?
40:44Well, we have to do anything that
40:45involves multiplication first. So, that
40:48-2 needs to be dealt with first. And
40:50actually, that's two transformations in
40:52there. The negative and the two. So, two
40:55things are going on. one we are
40:58reflecting
41:00over the x axis
41:03and then the two is we are vertically
41:07stretching by two.
41:10Now you can take care of these two at
41:12the same time cuz they both involve
41:13multiplication. You're just going to
41:15multiply y by -2.
41:19And then what else happens? while we
41:22shift left,
41:25right? Three units because of the plus
41:28three.
41:30Let me color code cuz you guys like
41:31that. Okay. And then these are the
41:34purple. And then four, we shift up one
41:37unit. That's outside, not touching the
41:39x. So, it's going to affect the y value.
41:41Shift up one unit. Pink. Okay. These two
41:45can happen in whatever order cuz they're
41:47just involving shifting addition. So you
41:50can do them same time, whatever. One at
41:52a time, live your life. So here we go.
41:57Um, how would I shift left three units?
41:59I would subtract three from X. And then
42:03how would I shift up one unit? I'd add
42:05one to Y. So what I tell my students
42:09when you're first starting off, you can
42:10just apply the transformations to the
42:12original key points. So first let's
42:16multiply y by -2.
42:21So it would become 0 0 1 -2
42:264 -4 right. And then we can subtract
42:30three from x and add 1 to y. So
42:32subtracting 3 from x that's -3 -2 1 and
42:37then add one to y 1 -1
42:42-3. So these are my final key points
42:44after all the transformations are
42:45applied. This is what we're going to
42:47graph. And then also just so your graph
42:49goes the right way, I always tell my
42:51students track what happened to 0 0
42:53because you don't have an arrow on that
42:55end point. You have an arrow on the
42:56other end. So here's 0 0 and it became
42:59-31. So don't put an arrow on this guy.
43:02Okay. So here, let's graph it. It's
43:03going to come out so beautiful.
43:05-31. I'm just thinking how
43:10Here we go. y
43:13X
43:17one two three 4 1 2 3 4
43:23Yes. It's kind of calming, right? To
43:26make your own little grid. -3 1. No
43:29arrow coming out of there. -21
43:331 -3. It's going like this. Like this.
43:39Oh yeah. Looks so good. Let's state the
43:41domain and range.
43:43Mhm. Domain is going to go from
43:47-3 to infinity, right? All the x values
43:50start at -3 to infinity. What about the
43:53rangeative
43:55infinity up until one? Yeah, perfect.
43:59Good job, you guys. Okay, let's [snorts]
44:02get a little fancier.
44:05Absolute value of a function. So to
44:07graph the absolute value of f ofx, start
44:10off just graphing f ofx. Then the
44:14portion of the graph for which f ofx is
44:16greater than or equal to zero would
44:18remain unchanged. It's already positive.
44:19It's already doing what it's supposed
44:20to. But the portion of the graph for
44:22which f ofx is less than zero gets
44:24reflected across the x-axis. So let's
44:27sketch the graph of f ofx= x cubed + 1.
44:31I'm going to do it step by step by step
44:32so you don't get all stressed out. So,
44:35first let's just graph y= x cubed. Okay.
44:40Hopefully you know what y= x cub looks
44:42like.
44:44So, here's my y ais. Here's my x axis.
44:49And it goes through 1 1 andgative
44:511gative 1. So, that's helpful. Here's
44:54one one. Oh, and 0 0 and 1 one negative
44:581 negative 1. Excuse me. So, it's like
45:00this.
45:02So, nice. Okay. Then inside, see how we
45:05have x cub + one. So shift the graph up
45:09one. So let's do that. y = x cubed +
45:13one.
45:15So here we go.
45:23Everybody goes up a unit. No problem.
45:30So now we have instead of 1 one 2 0 1
45:35and 1 0
45:38it's like this.
45:42Okay. Now I need the absolute value of
45:46this graph. So
45:48let me copy this so I don't have to do
45:50it all over.
45:53Whatever parts are already above the
45:55x-axis is fine.
45:58So if I want absolute value of this guy,
46:03this is fine. It's already greater than
46:06or equal to zero. This part has to get
46:09reflected above the x- axis. So it would
46:13look
46:15something like that. Okay? So then you
46:17would just come through. Yeah. Erase
46:20this guy. Get out of here. You're
46:22positive now. Quit being so negative.
46:25And that's our graph.
46:27See? And it was pretty painless.
46:30Okay. [clears throat]
46:32Now, let's talk about combinations of
46:34functions. So, you can combine functions
46:36using addition, subtraction,
46:37multiplication, division. And then we
46:40can take a composition of two functions.
46:44This one is a little trickier for
46:45students to deal with. So, given two
46:48functions f and g, the composite
46:50function f composed with g. So, you see
46:52how there's an open circle. It's not
46:54multiplication. Um, it's defined as
46:56follows. f composed with g ofx equals f
46:59of g ofx. So when you're getting used to
47:01this notation, I would immediately
47:03rewrite it this way. Whichever function
47:05comes second, that one's getting
47:06substituted into the other function. And
47:09be careful. Composition of functions is
47:11not commutative.
47:13I know. Okay. So let's let f ofx equal
47:16the square root of x. And g ofx is the
47:20cube root of 1 - x. Find f composed with
47:23g ofx and its domain.
47:26First things first, let's figure out
47:29what the domain is for each of them. So
47:31the domain of f ofx of x, it's going to
47:34be that x has to be greater than or
47:35equal to zero,
47:37right? You can't take the square root of
47:38a negative number. Now g of x is cube
47:40root of x. The index is odd. Three is an
47:44odd number. We can take odd roots of
47:47negative numbers. We totally can. So
47:49don't worry about that. G has no
47:52restrictions. Oh, its domain is all real
47:54numbers. Love that. Okay, so let's
47:57figure out the composition and then what
47:59the domain is for that new function. So
48:01f composed with g of x is equal to f of
48:06g of x like that. Okay, so it's f of
48:11cube root of 1 - x. Now, right here is
48:14where I always do a checkpoint and I go,
48:16do I have any domain restrictions here
48:18that I need to be aware of? because
48:19they're going to carry through. No, g
48:22had no restrictions on its domain. Okay,
48:24fantastic. Now f ofx tells me take the
48:27square root of whatever's in those
48:29parentheses. So I'm going to take the
48:30square root of the cube root of 1 - x.
48:35And you might say, "Oh me, oh my,
48:36Professor B, that looks wild." Well,
48:39here's what you do. You're going to have
48:40this quantity raised to the 16th. If you
48:44don't see it, I'll show you right now.
48:46Let's rewrite it using rational
48:48exponents. So cube root of 1 - x is 1 -
48:51x^ 1/3. Then if I take the square root,
48:54I'm raising it to the 1/2. Well, when
48:56you raise a power to another power, you
48:58multiply those exponents. So this is 1 -
49:01x^ 16, which is the 6th root of 1 - x.
49:07Oh wow. See, so the shortcut would have
49:10been Yeah. Just multiply these two
49:12indices and then boom, that's what
49:13you're left with. Okay, great. What's
49:16the domain of this function?
49:19Well, we're left with an even index. And
49:23remember, we can't take even roots of
49:25negative quantities. So that means we
49:27need 1 - x to be greater than or equal
49:30to zero, which means x has to be less
49:32than or equal to 1. So that's the domain
49:34for my new composition function. It's x
49:38such that x is less than or equal to 1.
49:41or if you want to write it in interval
49:42notation infinity to one.
49:46Okay, perfect. Now, in calculus, it's
49:49going to be frequently useful to be able
49:51to decompose functions. So, given g of x
49:55is equal to the square root of x over 1
49:58- x. Find two functions f and g such
50:01that g ofx is equal to f composed with g
50:04ofx. So, remember f composed with g ofx
50:07means f's your outer function. G is the
50:09function that's inside. So look here. Do
50:12you see two functions? One that's the
50:14outer function and then one that's
50:16inside. Yes. So this inside expression x
50:20over 1 - x. That's going to be g ofx.
50:22And then kind of ignore what's inside f
50:25ofx is just going to be roo<unk> of x.
50:28So this is f of to use our notation x
50:32over
50:33exactly 1 - x which becomes square<unk>
50:38of x over 1 - x. So we're going to say f
50:41ofx is<unk> x and g of x is x over 1 -
50:49x. This is going to be super useful when
50:51you learn your differentiation rules,
50:52chain rule, etc.
50:55Okay, very good. Let's look at another
50:57example here talking about rates. Now a
51:01spherical balloon is being inflated and
51:05the radius of the balloon is increasing
51:08at a rate of 2 cm per second. The reason
51:11why we want to talk about this is when
51:13you study calculus, you're going to
51:14study derivatives. And derivatives are
51:17used to measure the rate of change, the
51:20instantaneous rate of change of some
51:22quantity. And when we're looking at the
51:24rate of change with respect to time,
51:26it's very applicable, right, in so many
51:28physics, engineering problems, etc. So
51:32part A says, express the radius r of the
51:36balloon as a function of time. So right
51:39here, they told me the radius is
51:42increasing at a rate of 2 cm per second.
51:46So as t goes by every second the radius
51:50goes up by 2 cm 2 times however many
51:53seconds go by. So radius is a function
51:56of time that means I can write it as r
51:58of t. r is a function of t and it equals
52:002 t. t is in seconds.
52:04Write the volume of the balloon v as a
52:07function of t. Well they told me the
52:10balloon is a sphere. So yes, you are
52:12expected to know the formula for the
52:14volume of a sphere now that you're in
52:16calculus. Um these formulas are not
52:19handed to you. These are previous
52:21formulas, yes, from geometry and
52:22whatnot, but you have to have them in
52:24the forefront of your mind. So if you
52:26forgot, it's okay. I'll remind you right
52:27now. Volume of a sphere is 4/3 pi r
52:31cubed. But this is volume as a function
52:34of the radius. They want it as a
52:35function of t. So all I'm going to do is
52:38say no problem. I'm going to pop in
52:42how r is defined above. What is r
52:44defined as above? 2t. So now I can say,
52:48okay, I have v as a function of t. It's
52:504/3 pi * 8 t cubed. Let's clean that up.
52:56V of t is 32<unk>
53:00over 3
53:02t cubed. Perfect. Another formula that's
53:05going to come up a lot just in case you
53:07forgot. Surface area of a sphere.
53:12So surface area is equal to 4 pi r 2.
53:16What if I asked for that as a function
53:18of t? What would you do?
53:21Yeah, replace r again with 2t. This time
53:24it's just squared. That should make
53:26sense. Area is 2D, volumes 3D. So then
53:30we could say s of t = 4<unk> * 4 t^2.
53:35So s of t is 16<unk>
53:38t^2.
53:40Fantastic. Good job you guys.
53:43Okay,
53:45now we're going to move on. Shifting
53:47gears. As you know, for exponential
53:50functions, there's one base that's more
53:53useful or natural than any other, and
53:56this is the natural base e. And this
53:59base is especially user friendly for
54:01calculus students.
54:03And believe me, f ofx= e to x is going
54:06to become your favorite function. But it
54:10might seem kind of strange. You're like,
54:12e is approximately 2.72.
54:16I mean, that's as many as I really
54:18remember. Why is that natural? That
54:20doesn't seem like such a nice, beautiful
54:21number. But look at the graphs here of 2
54:24to the x and 3 to the x. And then also
54:28hopefully you can see the tangent lines
54:30at the point 01 are also shown below. So
54:33tangent to the curve is just touching or
54:35kissing the graph right there at that
54:37point 01. Two different tangent lines.
54:40Okay. Now notice the apparent slopes of
54:42these tangent lines. It turns out it's
54:45actually very useful to choose our base
54:48in such a way that the tangent line at
54:50the point 01 is one. And we don't quite
54:53have that here with either of these. But
54:56it turns out if E is the base and E is
54:58between two and three, right? E is about
55:012.72.
55:02That's the perfect base to pick so that
55:05the slope of this tangent line at 01 is
55:071. So make sure you're comfortable with
55:09the graph of e to the x. Be able to
55:12apply transformations to the graph etc
55:15etc because it's going to come up very
55:17extensively and frequently.
55:19All right, shifting gears. A function is
55:21one to one if it never takes on the same
55:23value twice. So stated algebraically f
55:26ofx1 is different from f ofx2 anytime x1
55:31is different from x2. So it's more
55:34precise than a function because a one
55:36one function different values of x never
55:39yield the same value of y. I just tell
55:42my students all the ordered pairs they
55:44are monogamous. Okay. Now one one
55:47functions can be identified graphically
55:49using the horizontal line test. So
55:52notice that you can use a vertical line
55:55test to first check if the graph is a
55:57function and then if so you can apply
55:58the horizontal line test afterwards to
56:00see if it's a one one function as well.
56:03If we know a function's always
56:06increasing
56:08or always decreasing then it has to be
56:10one one right? A horizontal line will
56:12never cross it more than once. Okay.
56:16Definition. Let f be a one:one function
56:20with domain a and range b. Then its
56:23inverse function denoted by f inverse.
56:28It's it looks like a little negative 1
56:29exponent, but it certainly is not an
56:31exponent. It has domain b and range a.
56:34We mentioned this a little bit earlier
56:36before. And it's defined as follows. f
56:38inverse of y is equal to x if and only
56:42if f ofx equals y for any y and b. But I
56:45want you to be so careful. So f inverse
56:49of x does not equal 1 over f ofx. This
56:54is not an exponent even though it looks
56:57like one. I'm so sorry I didn't come up
56:59with this notation. It is what it is. If
57:02you wanted to tell somebody, hey, take
57:04the reciprocal of f ofx, how would you
57:06write it? Let me show you. You would
57:09say, oh, you take f ofx and then you're
57:11going to raise it to the first power.
57:14That's how you would represent one over
57:15f ofx. Okay. But if that little negative
57:19one is next to the f mm- that means
57:21inverse.
57:23Okay. Perfect. Now notice that these
57:26definitions imply the cancellation
57:29equations that f inverse of f ofx= x for
57:32every x and a and that f of f inverse of
57:35x= x for every x and b. And often you're
57:39going to want to figure out what the
57:40inverse function is if you know that a
57:42certain function is one to one. So here
57:44are the steps, four steps. Replace f ofx
57:48with y if your function is a function of
57:51x. Interchange x and y. Solve the
57:55equation for the new y when possible and
57:57then rename it as f inverse. We'll do an
58:00example together in case you haven't
58:01done it for a minute. So given the
58:04following 1:1 function f ofx find f
58:07inverse of x. So first thing I'm going
58:09to do is replace f ofx with y. So we
58:13have y = 2x + 5 over 3x - 7. Step two,
58:19you're going to interchange all your x's
58:21and y's. So I'm going to replace that y
58:24with x and then those other x's become
58:27y's.
58:29Good.
58:31Okay. So far, pretty pretty relaxing.
58:33Step three, solve this equation for the
58:36new y if possible. Oh, it's possible.
58:38So, let's get the fractions out of
58:40there. I'm going to multiply through by
58:413 y - 7. So, 3 y - 7 * x = 2 y + 5. This
58:48is going to be 3xy - 7 x = 2 y + 5.
58:55Okay. How do I get the new y by itself?
58:58We'll collect all the terms that have y
59:00in it on the same side of the equation
59:02and then whoever doesn't have y gets
59:04banished to the other side. So yes,
59:07we'll have 3x y - 2 y over here and that
59:11equals 7 x + 5. And then we're almost
59:14there. The reason I want everybody with
59:16y on the same side is so I can factor it
59:19out. So, I can take y out and then I
59:22have 3x - 2
59:26= 7 x + 5.
59:29Divide by 3x - 2. And then we've
59:32isolated y. Look at that. That worked
59:33out beautifully. Wow. Wow. Wow. Over 3x
59:37- 2. And then last step, just put a nice
59:39little bow on it. Rename this y as f
59:42inverse of x. That way you know it's the
59:44inverse function of the one that was
59:46given to us. Right? That way it's
59:48crystal clear for whoever comes across
59:50it.
59:52Okay, very good. Let's box it with
59:55immense pride. Beautiful. Now,
59:58exponential functions are one one.
1:00:00Remember their graph. They're always
1:00:02increasing or always decreasing. That
1:00:04means we can take their inverse. You can
1:00:05only take the inverse of a function and
1:00:07have it be a function if it's one one.
1:00:10This leads us to the concept of a
1:00:12logarithm. So it means that if f ofx is
1:00:15equal to a raised to the x then the
1:00:17inverse function is equal to log base a
1:00:20of x where a log is defined as follows.
1:00:23So log base a of x equals y. This symbol
1:00:26means if and only if a raised to the y
1:00:30equals x. Um sometimes instead of like
1:00:33that double arrow you'll see if with
1:00:36like a couple fs written. That means if
1:00:38and only if.
1:00:40If and only if. So if and only if
1:00:43statements um mean that the forward and
1:00:45reverse statements are true
1:00:49that this way is true and the backwards
1:00:51way is true. Okay, good. So therefore
1:00:55the cancellation equations imply that
1:00:58log base a of a raised to the x is equal
1:01:00to x for every x that's in the domain.
1:01:04Every x in r and a raised to log base a
1:01:07of x equals x for every x that's greater
1:01:09than zero.
1:01:12Also, it's essential that you recall
1:01:15your properties of logarithms.
1:01:17There's only three here. Please, please
1:01:20don't be tempted to invent your own.
1:01:22That happens frequently. Then it's
1:01:24illegal. Students like to just go rogue.
1:01:27Please don't. Um, so if x and y are
1:01:29positive and r is any real number, then
1:01:31you can expand your logarithm as
1:01:33follows. If you have logarithm of a
1:01:34product, it's log base a of x plus log
1:01:36base a of y. You can read through this
1:01:38yourself. Okay?
1:01:40And then since we know that the natural
1:01:42base e is especially useful for
1:01:44exponential functions, it follows that
1:01:47this is a useful base for logarithms as
1:01:49well. And these are called natural
1:01:51logarithms. So anytime you have log base
1:01:54e of x, we write it as ln, natural log
1:01:57of x. And that means ln of x= y is
1:02:00equivalent to writing e to the y= x.
1:02:04Good. ln of e to the x is equal to x and
1:02:08e raised to the ln of x is equal to x so
1:02:10long as x is positive. It's really
1:02:13useful. Just remember that ln of e is
1:02:16equal to 1. That should make sense
1:02:18because ln remember is log base e of e.
1:02:22And what is a logarithm you guys? I know
1:02:23a lot of the times you just like use
1:02:25these words and you don't even have a
1:02:26good sense of what the heck they mean.
1:02:28What is a logarithm? A logarithm is an
1:02:30exponent. Yes, a logarithm is an
1:02:32exponent. Why am I saying that? Because
1:02:35this is asking me what's the exponent I
1:02:37put on e to get e one. Right? You raise
1:02:40it to the first power to get back
1:02:43itself. Exactly. And then look at our
1:02:46original definition here. Like we have
1:02:50this one. This will help you out. E to
1:02:53the y = x. That's equivalent to ln of x
1:02:57is equal to y. So that means my
1:03:00logarithm, my natural log is equal to y.
1:03:02Where was y in the exponential version?
1:03:05It was the exponent. So logarithms
1:03:10are exponents.
1:03:12Just for you to get a sense of what the
1:03:15heck are these things, right? Yes, I
1:03:17know. I know it all gets swimming in the
1:03:19brain. Oh, another thing very useful. I
1:03:22got off I got off track. ln of 1 is
1:03:24equal to zero. Again, what exponent do I
1:03:27put on e to get one? You have to raise
1:03:30it to the zero power. There you go.
1:03:33Okay, let's solve an exponential
1:03:35equation. You're going to certainly have
1:03:36to do this in calculus.
1:03:39Solve e 2x - 3 e x + 2 = z. The key is
1:03:44recognizing that this is quadratic in
1:03:46form. So e 2x I can write as e x^ 2 - 3
1:03:52* e x + 2 = 0. You can make a
1:03:56substitution or you can factor like the
1:03:58boss that you are. So this is going to
1:04:00be e x - 2 * e x -1 = 0. If you don't
1:04:06see that, right, you're like, I'm having
1:04:07a meltdown, Professor B, just let u= e
1:04:10x. Then this becomes u ^2 - 3 u + 2. And
1:04:15then hopefully you can see, oh yeah, now
1:04:18I can factor. So I just skipped that
1:04:20part. Okay, cool. From our zero product
1:04:23property, this tells me e to the x is 2
1:04:26or e to the x is 1.
1:04:29And then to solve for x, just take the
1:04:30natural log of both sides. So ln of e to
1:04:34x = ln of 2. Remember ln of e x just
1:04:38gives me back x
1:04:40or ln of e to x = ln of 1. So that means
1:04:45x is zero. So those both go in our
1:04:47solution set 0 and natural log of 2.
1:04:51Boom.
1:04:53Okay.
1:04:55Lastly, let's consider trigonometric
1:04:57functions.
1:04:58Let's think about y= sin of x. Is it 1
1:05:01one?
1:05:03Oh, okay. Do you know the graph of sinx?
1:05:06Let me jog your memory. So, its period
1:05:09is 2 pi. Its amplitude is one. Meaning
1:05:12the largest value it's ever going to
1:05:13take on is pos1 and the smallest value
1:05:16is 1. Okay. So, I'm just going to go up
1:05:18to one, down to 1. This is pi over 2
1:05:24pi
1:05:263 p<unk> over 2
1:05:29p<unk> over 2<unk>i
1:05:33-3<unk> / 2. It starts at 0 0 goes up to
1:05:38one at p<unk> / 2 back to 0 at pi etc.
1:05:42You should be comfortable like just
1:05:43plotting this general graph. Okay.
1:05:48So, oh my god, so wobbly. You know why?
1:05:50My hands all sweaty and tired. Sorry,
1:05:53guys.
1:05:56All right. Now, now, now, now. Here's
1:06:00the graph.
1:06:01Is it one one? So, it's a function. It
1:06:04passes the vertical line test. For it to
1:06:06be one one, it needs to also pass the
1:06:09horizontal line test. Does it?
1:06:12No, it certainly doesn't. Violates the
1:06:14horizontal line test. Look, intersects
1:06:16in multiple places. It actually will
1:06:19intersect an infinitely many because the
1:06:20graph continues.
1:06:23And these fools are asking me to find
1:06:25its inverse. I mean, come on. What we do
1:06:27is we pick a piece of the graph. We
1:06:30force it to be 1 one and we're going to
1:06:32restrict the domain. Yeah. And we
1:06:34restrict it to this piece right here.
1:06:37Why that piece? Well, because I'm going
1:06:39to attain all the y values from negative
1:06:411 to one, but it's just going to include
1:06:44that portion there where it would pass
1:06:46both the vertical and horizontal line
1:06:48test. So, let me tell you what's going
1:06:49on. To force this guy to force it to be
1:06:54one one,
1:06:56we restrict the domain.
1:07:01And just memorize. Yeah, this restricted
1:07:04domain is<unk> /2 to p<unk> /2. And that
1:07:08restricted domain is going to become the
1:07:10range on the inverse function. Super
1:07:13crucial. So now my x's are just
1:07:15between<unk>
1:07:17/ 2 and p<unk> / 2. And y is between -1
1:07:21and 1. So we can go ahead and find the
1:07:24inverse. Unfortunately, I need more
1:07:27paper. So allow me a moment to get some.
1:07:30Okay, so step one, look, we have y= sin
1:07:33of x. Cool. I'm finding the inverse
1:07:36function. I'm going to interchange x and
1:07:38y. So x= sin of y. Solve for the new y.
1:07:42Well, literally all you do is you just
1:07:44say, well, y is equal to sin inverse of
1:07:46x. Or alternatively, you could write y=
1:07:50r sin of x. Okay? And then lastly,
1:07:56[cough]
1:07:58replace y with f inverse of x. So you
1:08:00could say f inverse of x is equal to sin
1:08:03inverse of x or f inverse of x is equal
1:08:08to r sin of x. Okay. Now what is the
1:08:12domain and range of ark sign or sin
1:08:14inverse? This is crucial. So the domain
1:08:18is going to be -1 to 1. It's the range
1:08:20of what the original function was. And
1:08:21the range is restricted based on that
1:08:24restricted domain. It's only<unk> / 2 to
1:08:27p<unk> / 2. Keep this in mind when being
1:08:30asked to evaluate
1:08:33inverse trig functions. Okay, sine
1:08:36inverses range is restricted.
1:08:40Okay, perfect. So some notation the
1:08:43inverse sign function is denoted as
1:08:45follows. We went over that. This
1:08:46notation means that x is equal to s of y
1:08:48and y is between p<unk> /2 and p<unk> /
1:08:512. So in other words, y is the angle in
1:08:55p<unk> / 2 to p<unk> /2 whose sign is x.
1:08:57Anytime you're seeing an inverse trig
1:09:00function, it represents an angle. It
1:09:02represents an angle. So without using a
1:09:06calculator, evaluate sin inverse of 1 /
1:09:08rad 2. So you might be thinking, okay,
1:09:11on my unit circle, there's two places
1:09:13where s is equal to 1 rad 2. I know, but
1:09:16you can only give the value that's
1:09:18between<unk> /2 and p<unk> /2 when they
1:09:22ask for the value of an inverse trig
1:09:24function.
1:09:25So sin inverse of 1 over rad 2. We're
1:09:28going to get only the value in quadrant
1:09:291 which is pi over4.
1:09:33Okay, don't worry. I have a whole video
1:09:35on it if it's stressing you out. Okay,
1:09:37ark sign of - one2. So you have to ask
1:09:40yourself where is s=2.
1:09:43So here's the deal. Here's the deal,
1:09:46guys. Here's my unit circle. We know
1:09:49sign is negative in quadrants 1 and
1:09:52four, right?
1:09:54What am I saying? No, it's not in
1:09:56quadrants three and four. Oh me. Oh my.
1:09:59Here's x. Here's y. So where is sign
1:10:02equal to - one2?
1:10:04Well, it's going to be down here or down
1:10:07here. Now you're not even looking over
1:10:09here. Pretend that doesn't happen
1:10:11because remember we're only going to get
1:10:13values that are between<unk> / 2 and
1:10:16p<unk> / 2. That's there. This is pi /
1:10:192. This is p<unk> / 2. So what angle
1:10:25Mhm. has a sign of - 1/2<unk> / 6.
1:10:30And you might say, "Oh, Professor B, can
1:10:31I say 11<unk> / 6 because that's
1:10:33co-terminal?" No, because 11<unk> over 6
1:10:36means you went this way and you went out
1:10:40of bounds. You have to stay between
1:10:42p<unk> /2 and p<unk> /2. Oh yeah. Yeah.
1:10:45You got to focus when you do these.
1:10:47Okay.
1:10:49You should also recall that to graph f
1:10:51ofx= sin inverse ofx, we're going to use
1:10:54the fact that the graph of an inverse
1:10:55function is obtained by reflecting the
1:10:58graph of the original function across
1:11:00the line y= x. And so we're going to use
1:11:02a lot of inverse sign, inverse cosine,
1:11:05inverse tangent functions this semester.
1:11:07And as you've seen in previous courses,
1:11:10you get these functions using similar
1:11:12methods. So inverse cosine function is
1:11:15denoted y= cosine inverse of x or arc
1:11:17cos ofx and this means x is equal to
1:11:21cosine of y and y is between 0 and pi.
1:11:26Why? Because we had to restrict the
1:11:27domain again on cosine to make it one
1:11:29one to find this inverse just like we
1:11:31did with s. But because cosine's graph
1:11:33is shifted a little differently we
1:11:35restrict it to a different portion. So
1:11:37cosine's graph, yeah, it's between 1 and
1:11:40-1,
1:11:41but here's pi, pi / 2, here's 2 pi.
1:11:45Cosine's graph starts up here, goes
1:11:47down, then back up. So because cosine's
1:11:51graph looks like this, we can't restrict
1:11:53it from p<unk> / 2 to p<unk> / 2 like we
1:11:55did with s. It wouldn't be 1 one right
1:11:58there. We have to cut it off from 0 to
1:12:00pi. That's why we get that different
1:12:02restriction. Okay? And then inverse
1:12:05tangent function is denoted y= tan
1:12:07inverse of x or arc tan of x. And
1:12:11similarly we just take the portion
1:12:14between<unk> / 2 and p<unk> / 2. So
1:12:17tangent's graph
1:12:20repeats its period is just pi not 2 pi
1:12:23like s and cosine. Okay. And then here's
1:12:27pi / 2. Here's p<unk> / 2. And we just
1:12:32take the part this one period of tangent
1:12:34because it is one to one right here. Oh
1:12:37yeah. And because it has those
1:12:39asmmptotes, we're not going to include
1:12:40the end points. That's why we have
1:12:41parentheses there. Okay. Um the other
1:12:45inverse trig functions are less
1:12:47frequently used. But I do want to say
1:12:50before we wrap things up, oh my god,
1:12:52we've been having so much fun. I can't
1:12:54believe it's over. that in calc 2
1:12:58especially
1:13:00now as well but in calc 2 especially
1:13:04you're going to need that graph of tan
1:13:06inverse of x also burned in the brain or
1:13:08tattooed on you. Oh yeah let's graph it
1:13:11together right now. So remember when you
1:13:15find the inverse function you
1:13:16interchange x's and y's. So the graph
1:13:19you can obtain very similarly
1:13:20interchange all the x's and y's your
1:13:22ordered pairs.
1:13:24Similarly, your vertical asmmptotes
1:13:26become horizontal asmmptotes.
1:13:29So here's y = tan inverse of x coming in
1:13:31hot. Instead of vertical asmmptotes at
1:13:34p<unk> / 2 and -2, that's right. I've
1:13:36got horizontal asmmptotes at pi / 2
1:13:39and<unk> / 2
1:13:43right here. I know there's a dash line
1:13:46feature, but I can't be bothered. Here's
1:13:48my asmtote. Here's my other asmtote. And
1:13:51then it looks like someone came and kind
1:13:53of knocked it over on its side
1:13:56like this. This is y= tan inverse of x.
1:13:59And then when you're in calculus, you're
1:14:01going to be asked things like what's the
1:14:02limit as x approaches infinity of tan
1:14:05inverse of x? And you're just going to
1:14:07have to imagine the graph to get the
1:14:08answer. The answer is pi / 2. Or what's
1:14:12the limit as x approaches infinity of
1:14:15tan inverse of x? Yeah, you're going to
1:14:18imagine the graph and you're going to
1:14:20tell everybody with immense confidence
1:14:21while the rest of your classmates are
1:14:23like, "Huh, what's going on?" It's pi /
1:14:262. So, make sure you remember this
1:14:28graph.
1:14:30Okay, that wraps up our essential
1:14:32algebra and trig review for calc 1. Now,
1:14:35if some of the topics felt a little
1:14:36rusty, that's exactly what a review like
1:14:38this is for. So, go back to the sections
1:14:40that you need, work through the examples
1:14:42on your own, and make sure that you're
1:14:44comfortable with these skills as you
1:14:46move into calculus. And remember, my
1:14:48entire calculus 1 lecture library is
1:14:50available for free right here on
1:14:52YouTube. If you're looking for a more
1:14:53structured learning experience, head to
1:14:55professorvmath.com.
1:14:57My complete self-paced calculus 1 course
1:15:00is available. It includes the
1:15:02downloadable lecture template that I
1:15:04worked through in this video along with
1:15:06guided notes, problem sets, study
1:15:08guides, sample exams, and the best part
1:15:10is that you get lifetime access. Thanks
1:15:13so much for watching and I'll see you in
1:15:15the next video. Don't forget to give
1:15:16this video a thumbs up, subscribe if you
1:15:18haven't already, and you can also follow
1:15:20me on Instagram and Tik Tok, Math with
1:15:22Professor V. Bye, guys.

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