Ex: Decompose Functions — Transcript

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1. 0:00Welcome to three examples of expressing a function  as a composite function. Your directions may
2. 0:06also say "decompose a given function." Here,  we’re given h of x equals the quantity two x
3. 0:11plus three raised to the third power. We want to  write h of x as a composite function in the form
4. 0:17f of g of x. Remember, we can also  think of this as f after g of x.
5. 0:23When decomposing a function, I think it's  often helpful to use the alternative form of
6. 0:28a composite function given by this equation here.  So we want h of x equal to f of g of x, which we
7. 0:40can also write in this form here: f of g of x. The  reason this form is helpful is that, in this form,
8. 0:48it's easy to see that g of x is the inner function  and function f would be the outer function.
9. 0:54So when decomposing a function, we want  to be able to look at the given function
10. 0:58and determine which part is the inner  function and which part is the outer function.
11. 1:02Looking at h of x, our inner function is  going to be the quantity two x plus three,
12. 1:08and our outer function, function f,  is going to be the cubing function.
13. 1:14So to decompose a function in the form of f of g  of x, we want to define function g and function f
14. 1:20so that this composite function is equal to the  original function. So our ultimate goal is to
15. 1:26find what we're going to use for function f of x  and function g of x. But we already said g of x is
16. 1:33the inner function, so in this case, it would be  the quantity two x plus three, and our function f,
17. 1:40the outer function, is going to be the cubing  function. So f of x is equal to x cubed.
18. 1:46Now, function decomposition is not unique,  but to me, this seems like the most obvious
19. 1:51decomposition. So these would be our two  functions, so that f of g of x is equal to h of x.
20. 1:59And let's go ahead and check this. h of x
21. 2:02should be equal to f of g of x. Let's go and  write it in this form here. Well, g of x is
22. 2:10equal to the quantity two x plus three, so we can  write this as f of the quantity two x plus three.
23. 2:18So this quantity becomes the input  into function f, which cubes its
24. 2:22input. So this does give us h of x, which is the  quantity two x plus three raised to the third.
25. 2:30So we've decomposed this function correctly.
26. 2:33Let's take a look at a second example.
27. 2:37Same question, different function. So  we want h of x equal to f of g of x,
28. 2:46which we'll go and write using the  alternative form or this form here.
29. 2:53So again, g of x will be our inner function, and  f will be our outer function. Looking at h of x,
30. 3:00let's let our inner function g of x equal the  linear function five x minus one. And if we do
31. 3:08this, then our function f, the outer function,  will be equal to the square root function.
32. 3:14So again, to decompose function h into f of  g of x, our goal is to define function f of
33. 3:21x and define function g of x. As we already  said, g of x is the quantity five x minus one,
34. 3:29our linear function, and f of x will be the  square root function, or the square root of x.
35. 3:36This is what we're being asked to find,  and let's go ahead and check this,
36. 3:40meaning we want to make sure h  of x is equal to f of g of x.
37. 3:47Well, g of x is equal to the quantity five x minus  one, so this is equal to f of five x minus one.
38. 3:55This quantity becomes the input  into function f, so we replace
39. 3:59x with five x minus one, giving us  the square root of five x minus one,
40. 4:05which is h of x. So that checks as well,  and again, these functions are not unique,
41. 4:10but to me, this does seem like the most  obvious composition for function h.
42. 4:15Let's try one more. We have h of x equals  one divided by the quantity x minus five,
43. 4:21and the question is the same: we want to write  h of x as the composite function f of g of x,
44. 4:31which we can also write using this notation here.
45. 4:37So we need to identify the inner function and  outer function. This one may not be quite as
46. 4:41obvious. g of x is our inner function. Let's let g  of x equal the denominator of a rational function;
47. 4:49therefore, the outer function f is just going to  be the rational function one divided by its input.
48. 4:58So again, we have f of x, and we have g of x,
49. 5:03and we're saying g of x will  equal the quantity x minus five,
50. 5:08and therefore the outer function will be this  fraction function, which would just be one over
51. 5:12x. So these are the two functions the  question is asking us to determine.
52. 5:20Let's go and check this to make sure our  composite function is equal to h of x.
53. 5:31Now, we'll replace g of x with x minus  five, so we'll have f of x minus five.
54. 5:38This becomes the input into function f,  which is just one divided by its input or
55. 5:43one divided by, in this case, x  minus five, which is equal to h of x.
56. 5:51And that's going to do it for this  lesson. I hope this was helpful.