Ex 1: Composite Function Values — Transcript

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1. 0:00We want to determine the composite function values  f of g of zero and g of f of zero, given f of x
2. 0:07equals twelve plus three x minus x squared and  g of x equals negative three x plus fifteen.
3. 0:14By definition, f of g is equal to f of g  of x, so f of g of zero is equal to f of
4. 0:24g of zero. To evaluate this, we'll  start with the innermost function,
5. 0:29which means we'll find g of zero first.  So let's go ahead and do that below.
6. 0:36Since g of x is equal to  negative three x plus fifteen,
7. 0:41g of zero is going to be equal to  negative three times zero plus fifteen.
8. 0:48So, g of zero is equal to fifteen. Now  we'll substitute fifteen for g of zero.
9. 0:58So our composite function is equal to f of
10. 1:03fifteen. Now we'll substitute fifteen into f  to determine this composite function value.
11. 1:09So we'd have twelve plus three  times fifteen minus fifteen squared.
12. 1:19So we'll have twelve plus this will be forty-five  minus fifteen squared, which is equal to two
13. 1:28hundred twenty-five. Twelve plus forty-five  is equal to fifty-seven. Fifty-seven minus
14. 1:32two hundred twenty-five is equal to  negative one hundred sixty-eight.
15. 1:39So, f of g of zero
16. 1:45is equal to negative one hundred sixty-eight.
17. 1:54And for the second example, since  g of f is equal to g of f of x,
18. 2:02g of f of zero is equal to g of f of zero.
19. 2:09Again, starting with the innermost  function, we'll start by determining
20. 2:13f of zero. Let's go ahead and do  that below. Since f of x is equal to
21. 2:21twelve plus three x minus x squared, we'd have  twelve plus three times zero minus zero squared.
22. 2:32So, f of zero is equal to twelve. And  since f of zero is equal to twelve,
23. 2:41this simplifies to g of twelve.
24. 2:44Now we'll substitute twelve into g, where g  is equal to negative three x plus fifteen.
25. 2:51So we'll have negative three  times twelve plus fifteen.
26. 2:57Well, that's going to be negative thirty-six plus  fifteen, which is equal to negative twenty-one.
27. 3:04So, g of f of zero is equal  to negative twenty-one.