Key Concept 1

1. Key Concept 1

2. The domain of f and g are both so the domain of (f + g) is Answer: Operations with Functions A. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum oftwo functions = (x2 – 2x) + (3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = x2 + x – 4 Simplify. Example 1

3. The domain of f and h are both so the domain of (f – h) is Answer: Operations with Functions B. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x2 – 2x) – (–2x2 + 1) f(x) = x2 – 2x; h(x) = –2x2 + 1 = 3x2 – 2x – 1 Simplify. Example 1

4. The domain of f and g are both so the domain of (f ● g) is Answer: Operations with Functions C. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f(x) ● g(x) Definition of product of two functions = (x2 – 2x)(3x – 4) f(x) = x2 – 2x; g(x) = 3x – 4 = 3x3 – 10x2 + 8x Simplify. Example 1

5. D. Given f(x) = x2 – 2x, g(x) = 3x – 4, and h(x) = –2x2 + 1, find the function and domain for Operations with Functions Definition of quotient of two functions f(x) = x2 – 2x; h(x) = –2x2 + 1 Example 1

6. The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of . So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Answer: D = (–∞, 0) È (0, 2) È (2, ∞) Operations with Functions Example 1

7. Find (f + g)(x), (f – g)(x), (f ● g)(x), and for f(x) = x2 + x, g(x) = x – 3. State the domain of each new function. Example 1

8. A. B. C. D. Example 1

9. Key Concept 2

10. = 2(x2 + 6x + 9) – 1 Expand (x +3)2 = 2x2 + 12x + 17 Simplify. = f(x + 3) Replace g(x) with x + 3 = 2(x + 3)2 – 1 Substitute x + 3 for x in f(x). Compose Two Functions A. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](x). Answer: [f ○ g](x) = 2x2 + 12x + 17 Example 2

11. = (2x2 – 1) + 3 = 2x2 + 2 Substitute 2x2 – 1 for x in g(x). Simplify Compose Two Functions B. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [g ○ f](x). Answer: [g ○ f](x) = 2x2 + 2 Example 2

12. Compose Two Functions C. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: [f ○ g](2) = 49 Example 2

13. Find for f(x) = 2x – 3 and g(x) = 4 + x2. A. 2x2 + 11; 4x2 – 12x + 13; 23 B. 2x2 + 11; 4x2 – 12x + 5; 23 C. 2x2 + 5; 4x2 – 12x + 5; 23 D. 2x2 + 5; 4x2 – 12x + 13; 23 Example 2

14. A. Find . Find a Composite Function with a Restricted Domain Example 3

15. To find , you must first be able to find g(x) = (x – 1)2, which can be done for all real numbers. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x) > 1. Excluding from the domain those values for which 0 < (x – 1)2 <1, namely when 0 < x < 1, the domain of f ○ g is (–∞, 0] È [2, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

16. Notice that is not defined for 0 < x < 2. Because the implied domain is the same as the domain determined by considering the domains of f and g, we can write the composition as for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Replace g(x) with (x – 1)2. Substitute (x – 1)2 for x in f(x). Simplify. Example 3

17. Answer: for (–∞, 0] È [2, ∞). Find a Composite Function with a Restricted Domain Example 3

18. B. Find f ○ g. Find a Composite Function with a Restricted Domain Example 3

19. To find f ○ g, you must first be able to find , which can be done for all real numbers x such that x2 1. Then you must be able to evaluate for each of these g(x)-values, which can only be done when g(x)  0. Excluding from the domain those values for which 0 >x2 – 1, namely when –1 < x< 1, the domain of f ○ g is (–∞, –1) È (1, ∞). Now find [f ○ g](x). Find a Composite Function with a Restricted Domain Example 3

20. Find a Composite Function with a Restricted Domain Example 3

21. Answer: Find a Composite Function with a Restricted Domain Example 3

22. Check Use a graphing calculator to check this result. Enter the function as . The graph appears to have asymptotes at x = –1 and x = 1. Use the TRACE feature to help determine that the domain of the composite function does not include any values in the interval [–1, 1]. Find a Composite Function with a Restricted Domain Example 3

23. Find a Composite Function with a Restricted Domain Example 3

24. Find f ○ g. A. D =(–∞, –1)  (–1, 1)  (1, ∞); B. D =[–1, 1]; C. D =(–∞, –1)  (–1, 1)  (1, ∞); D. D =(0, 1); Example 3

25. A. Find two functions f and g such that when . Neither function may be the identity function f(x) = x. Decompose a Composite Function Example 4

26. h Sample answer: Decompose a Composite Function Example 4

27. B. Find two functions f and g such that when h(x) = 3x2 – 12x + 12. Neither function may be the identity function f(x) = x. Decompose a Composite Function h(x) = 3x2 – 12x + 12 Notice that h is factorable. = 3(x2 – 4x + 4) or 3(x – 2)2 Factor. One way to write h(x) as a composition is to let f(x) = 3x2 and g(x) = x – 2. Example 4

28. Decompose a Composite Function Sample answer:g(x) = x – 2 and f(x) = 3x2 Example 4

29. A. B. C. D. Example 4