Composite Functions — Transcript

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1. 0:00Welcome to a video on composite functions.  A composite function can be thought of as
2. 0:06a combination of two functions. One way to  illustrate this would be to use a conveyor belt,
3. 0:12where we have an initial input, let's say three.  One way to illustrate this would be to use a
4. 0:17conveyor belt, where we have an initial input,  let's say three, and it becomes the input into f,
5. 0:23where f squares its input and produces an output  of nine. This output becomes the input into g, so
6. 0:31we multiply two times nine, and the final output  of this composite function would be eighteen.
7. 0:39Now, the order in which we evaluate a composite  function is extremely important. For example, if
8. 0:44we have the same functions but now we reverse the  order and we start with the same input of three,
9. 0:51function g would multiply that by two, producing  an output of six, which becomes the input into f,
10. 0:57and six squared would be thirty-six. So notice, if  we change the order in which we compose functions,
11. 1:03the outputs in most cases will be quite  different, even though our inputs may be the same.
12. 1:11The formal definition of a composite function  states that given two functions f and g, the
13. 1:17composite function, denoted using this notation,  is read "f composed with g" or just "f of g."
14. 1:27We need to be careful not to confuse this symbol  here with a multiplication sign. Notice for a
15. 1:32composition of functions, it's a hollow circle,  whereas for multiplication, it would be solid.
16. 1:38Below, we see three equivalent ways to  represent a composition of functions. Remember,
17. 1:44for a composite function, the order in which we  compose the functions is extremely important.
18. 1:50Therefore, when we're given a composite function,  we're going to rewrite it in this form here,
19. 1:55because this tells us that  we're going to first evaluate
20. 1:59function g at the value of x, and the output  of that will be the input into function f.
21. 2:05So we're going to evaluate function g first and  then function f. If the composition of functions
22. 2:12is written in this way, and that's almost in  the opposite direction that you might think.
23. 2:17Notice the f comes before the g, but when  it comes to evaluating the given value of x,
24. 2:22we'll first be the input into g, and  that output will become the input into f.
25. 2:28Let's go and look at some examples. If  we want to evaluate f of g of three,
26. 2:33this is going to be equal to f of g of three  written this way. So this tells us that we
27. 2:40will find the value of g of three first,  and that result will be the input into f.
28. 2:47So let's go down here on the  side and find g of three.
29. 2:52g of three will equal three cubed minus five.
30. 2:56Well, that'll be twenty-seven minus five, which is  equal to twenty-two. Since g of three is equal to
31. 3:02twenty-two, this just becomes f of twenty-two. We  found g of three, and it was equal to twenty-two.
32. 3:11Now, twenty-two will become the input into  f. Notice here f multiplies the input by two
33. 3:18and then subtracts one, so we have  forty-four minus one. So we have forty-three.
34. 3:25So f of g of three is equal to forty-three. Let's  go and take a look at another one. Notice this
35. 3:35question has the same input, but now instead of f  of g, it's g of f. So this is equal to g of f of
36. 3:42three. So now we're going to find f of  three first. Let's go down here and do that.
37. 3:48f of three is equal to two times x  minus one or two times three minus one.
38. 3:55That'll be six minus one or five, so f of three is  equal to five. So we can replace this with five.
39. 4:03So this becomes g of five, and g of x is equal  to x cubed minus five. So we'll have five
40. 4:10cubed minus five. So that's one  hundred twenty-five minus five
41. 4:15is equal to one hundred twenty. So this  would be the value of g of f of three.
42. 4:24Let's try another. We want h of g of  negative one. Well, this is equal to
43. 4:32h of g of negative one written this way, and this  should key us in to evaluate g of negative one
44. 4:39first. So we'll go ahead and do this on  the side. g of negative one, well, g of x
45. 4:47is equal to x cubed minus five, so we have  negative one cubed minus five. Well, this
46. 4:52will be negative one minus five or negative six.  So g of negative one is equal to negative six.
47. 4:58So now we can rewrite this as h  of negative six. So now we replace
48. 5:04x with negative six in h, so we'll have five minus  negative six squared. Watching our signs here,
49. 5:11we're going to have five minus a positive  thirty-six, which is equal to negative thirty-one.
50. 5:18So h of g of negative one is  equal to negative thirty-one.
51. 5:26Okay, now there's one other type of problem  we need to look at. Notice on this problem
52. 5:30they don't give us an initial input. So what  we're going to have to do is input one of the
53. 5:36functions into the other function, and again,  it's important that we follow the correct order.
54. 5:41So remember this is the same as f of g of x, which  by definition is equal to f of g of x written this
55. 5:47way. Notice there's no x value to input into g,  so the only we can really do is replace g of x
56. 5:56with x squared minus x plus five. So we need  to determine f of x squared minus x plus five.
57. 6:04So if there's no value to input into g, we have  to replace g of x with the entire function.
58. 6:10So now wherever we see an x in f, we'll replace  it with this entire expression. So instead of
59. 6:15four x plus one, we'll have four times the  quantity x squared minus x plus five plus one.
60. 6:23So we'll distribute the four and then  combine any like terms, so four x squared
61. 6:29minus four x plus twenty plus one. So f of g
62. 6:36is equal to four x squared minus four  x, and we'd have plus twenty-one.
63. 6:44Sometimes these are a little tricky when you  first do these, but the process is exactly
64. 6:48the same. Except instead of having a numerical  value, you have to substitute an entire function
65. 6:53into the other function. Let's try one more of  these. Now we have g of f. Instead of f of g,
66. 7:00so this is equal to g of f of x. Again, we  don't have an input to put into function
67. 7:07f, so we're going to rewrite this as g  of f of x is equal to four x plus one.
68. 7:15So wherever we see an x in g, we'll replace it  with four x plus one. So we're going to have,
69. 7:21instead of x squared, we'll have four x plus  one squared minus four x plus one plus five.
70. 7:32Here we have to square four x plus one. There's  no shortcuts here; we're going to have four x
71. 7:37plus one times four x plus one. This  will be minus four x minus one plus five.
72. 7:46Okay, so we're going to have sixteen x squared  plus four x plus four x, that's plus eight x
73. 7:54plus one minus four x. This would  be plus four, and one last step.
74. 8:05You can see this type of problem is a lot more  work. We're going to have sixteen x squared
75. 8:12plus four x. Here we have a positive  one and a positive four plus five.
76. 8:20And notice that when we change the order  of the composition, the result was quite
77. 8:24different from what we had on the previous  screen. One last comment I'd like to make:
78. 8:28if we did determine f of g and it was equal to g  of f, which was equal to x, then the two functions
79. 8:35have a special relationship, and they are inverses  of one another. This topic will be covered
80. 8:41in another video, so thank you  for watching, and have a good day.