Ex: Evaluate Composite Functions from Graphs — Transcript

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1. 0:00We want to determine the composite function  values from the graphs of function f in blue
2. 0:07and function g in red. From this definition  here, we see there are two ways to express a
3. 0:13composite function. I think it's going to be  helpful to rewrite these composite functions
4. 0:18using the notation given here on the right.
5. 0:21For f of g of 1, or f after g of 1, we  can write this using this notation here.
6. 0:31For g of f of 9, or g after f of  9, we can write this notation here.
7. 0:42For f of f of 0, we can write like this.
8. 0:49And for g of g of 8, we can write like this.
9. 0:55The important thing to remember  when evaluating a composite function
10. 0:59is to know which function to evaluate  first. Using this notation here,
11. 1:03I think it's easier to see we're always  going to evaluate the inner function first.
12. 1:07So for this composite function,  we'll determine the value of g of 1
13. 1:11first. Another way to remember this  would be to read this as f after g of 1.
14. 1:18To determine the value of g of 1, we need to  find the y-coordinate on g when x equals 1.
15. 1:25So here's the graph of g, and here's x equals 1.
16. 1:29We go straight to the function. When x  is 1, notice the corresponding y value
17. 1:34or function value is 8. Since g of 1  equals 8, we can write this as f of 8.
18. 1:43Now we'll go to the graph of function f and  determine the y-coordinate when x equals 8.
19. 1:50So here's the graph of f, and here's x equals 8.  When x is 8, notice the corresponding y value is
20. 2:005. So f of 8 equals 5. Thus,  f of g of 1 is equal to 5.
21. 2:14Next, we have g of f of 9, so we'll start by  evaluating f of 9. We go to the graph of function
22. 2:21f and determine the y-coordinate when x is 9.  Here's the graph of f, and here's x equals 9.
23. 2:29The corresponding y value or function value is 3.  Since f of 9 equals 3, this simplifies to g of 3.
24. 2:41Now we'll go to the graph of function g  and determine the y value when x is 3.
25. 2:47Here's x equals 3, and the corresponding  y value is 6, which is our function value.
26. 2:56So this equals 6. Thus, g of f of 9 is equal to 6.
27. 3:05Let's take a look at our last two examples. We  have f of f of 0, and we'll start by determining
28. 3:10the value of f of 0. We find the y-coordinate  when x equals 0 for function f. Notice when x
29. 3:18equals 0, we would be right here on the  y-intercept. The corresponding y value is 1.
30. 3:27Since f of 0 equals 1, this simplifies to f  of 1. Using the same graph, we'll find the
31. 3:34y-coordinate when x equals 1. So here's x  equals 1, and the corresponding y value is 2.
32. 3:43This equals 2, which means  f of f of 0 is equal to 2.
33. 3:52And for our last example, we have g of  g of 8. Using the graph of function g,
34. 3:58we'll determine the y-coordinate when x  equals 8 to evaluate g of 8. When x equals 8,
35. 4:06the function value is 2,  so this is equal to g of 2.
36. 4:16Then using the graph of function g again, we'll  find the y-coordinate when x equals 2. When x is
37. 4:242, the y-coordinate or corresponding function  value is 7. This is equal to 7. Therefore, g
38. 4:36of g of 8 is equal to 7.
39. 4:42That'll do it for this example.  I hope you found this helpful.