1. Key Concept 1

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

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

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

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

6. Example 1 Operations with Functions 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)  (0, 2)  (2, ∞). Answer: D = (–∞, 0)  (0, 2)  (2, ∞)

7. Example 1 Find (f + g)(x), (f – g)(x), (f ● g)(x), and for f (x) = x 2 + x, g (x) = x – 3. State the domain of each new function.

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

9. Key Concept 2

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

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

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

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

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

15. Example 3 Find a Composite Function with a Restricted Domain 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]  [2, ∞). Now find [f ○ g](x).

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]  [2, ∞). Example 3 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.

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

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

19. Example 3 Find a Composite Function with a Restricted Domain To find f ○ g, you must first be able to find, which can be done for all real numbers x such that x 2  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 > x 2 – 1, namely when –1 < x < 1, the domain of f ○ g is (–∞, –1)  (1, ∞). Now find [f ○ g](x).

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

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

22. Example 3 Find a Composite Function with a Restricted Domain 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].

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

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

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

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

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

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

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