1. A. The parent graph is translated left 3 units and up 4 units.
Use the graph of y = x 2 to describe the graph of the related function y = (x – 4)2 – 3.
A. The parent graph is translated left 3 units and up 4 units.
B. The parent graph is translated right 3 units and down 4 units.
C. The parent graph is translated left 4 units and down 3 units.
D. The parent graph is translated right 4 units and down 3 units.
5–Minute Check 2

2. A. The parent graph is translated left 3 units and up 4 units.
Use the graph of y = x 2 to describe the graph of the related function y = (x – 4)2 – 3.
A. The parent graph is translated left 3 units and up 4 units.
B. The parent graph is translated right 3 units and down 4 units.
C. The parent graph is translated left 4 units and down 3 units.
D. The parent graph is translated right 4 units and down 3 units.
5–Minute Check 2

3. You evaluated functions. (Lesson 1-1)
Perform operations with functions.
Find compositions of functions.
Then/Now

4. Key Concept 1

5. (f + g)(x) = f(x) + g(x) Definition of sum of two functions
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 – 4 Simplify.
The domain of f and g are both so the domain of (f + g) is
Answer:
Example 1

6. (f + g)(x) = f(x) + g(x) Definition of sum of two functions
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 – 4 Simplify.
The domain of f and g are both so the domain of (f + g) is
Answer:
Example 1

7. (f – h)(x) = f(x) – h(x) Definition of difference of two functions
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 – 1 Simplify.
The domain of f and h are both so the domain of (f – h) is
Answer:
Example 1

8. (f – h)(x) = f(x) – h(x) Definition of difference of two functions
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 – 1 Simplify.
The domain of f and h are both so the domain of (f – h) is
Answer:
Example 1

9. (f ● g)(x) = f (x) ● g(x) Definition of product of two functions
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 + 8x Simplify.
The domain of f and g are both so the domain of (f ● g) is
Answer:
Example 1

10. (f ● g)(x) = f (x) ● g(x) Definition of product of two functions
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 + 8x Simplify.
The domain of f and g are both so the domain of (f ● g) is
Answer:
Example 1

11. Definition of quotient of two functions
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
Example 1

12. 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) È (0, 2) È (2, ∞).
Example 1

13. 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.
Example 1

14. A. B. C. D.
Example 1

15. A. B. C. D.
Example 1

16. Key Concept 2

17. A. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [f ○ g](x).
Compose Two Functions
A. Given f (x) = 2x2 – 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
Expand (x +3)2
= 2(x 2 + 6x + 9) – 1
Simplify.
= 2x x + 17
Example 2

18. B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x).
Compose Two Functions
B. Given f (x) = 2x2 – 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
Example 2

19. C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2).
Compose Two Functions
C. Given f (x) = 2x 2 – 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.
Example 2

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

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

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

23. 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] È [2, ∞). Now find [f ○ g](x).
Example 3

24. Substitute (x – 1)2 for x in f (x).
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.
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, ∞).
Example 3

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

26. 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 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 > x 2 – 1, namely when –1 < x < 1, the domain of f ○ g is (–∞, –1) È (1, ∞). Now find [f ○ g](x).
Example 3

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

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

29. 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].
Example 3

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

31. 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

32. 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

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

34. Decompose a Composite Functionh
Example 4

35. Decompose a Composite Functionh
Sample answer:
Example 4

36. B. Find two functions f and g such that
Decompose a Composite Function
B. Find two functions f and g such that
when h (x) = 3x 2 – 12x Neither function may be the identity function f (x) = x.
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

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

38. A.
B.

C.

D.

Example 4

39. A.
B.

C.

D.

Example 4

40. So, the functions are R(t) = 25 + 10t and A(R) = R 2.
Compose Real-World Functions
A. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find functions to model the data.
The length r of the radius increases at a rate of 10 pixels per second, so R(t) = t, where t is the time in seconds and t  0. The area of the circle is  times the square of the radius. So, the area of the circle is A(R) = R 2.
So, the functions are R(t) = t and A(R) = R 2.
Example 5

41. A ○ R = A[R(t)] Definition of A ○ R
Compose Real-World Functions
B. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find A ○ R. What does the function represent?
A ○ R = A[R(t)] Definition of A ○ R
=A( t) Replace R (t) with t.
= ( t)2 Substitute ( t) for R in A(R).
= 100t t + 625 Simplify.
Example 5

42. Compose Real-World Functions
So, A ○ R = 100t t + 625. The composite function models the area of the circle as a function of time.
Example 5

43. BUSINESS A satellite television company offers a 20% discount on the installation of any satellite television system. The company also advertises $50 in coupons for the cost of equipment. Find [c ◦ d](x) and [d ◦ c](x). Which composition of the coupon and discount results in the lower price? Explain.
A. [c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer: [d ◦ c](x) represents the cost of installation using the coupon and then the discount results in the lower cost.
B. [c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer: [c ◦ d](x) represents the cost of installation using the discount and then the coupon results in the lower cost.
C. [c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer: [c ◦ d](x) represents the cost of installation using the discount and then the coupon results in the lower cost.
D. [c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer: [c ◦ d](x) represents the cost of installation using the coupon and then the discount results in the lower cost.
Example 5

44. BUSINESS A satellite television company offers a 20% discount on the installation of any satellite television system. The company also advertises $50 in coupons for the cost of equipment. Find [c ◦ d](x) and [d ◦ c](x). Which composition of the coupon and discount results in the lower price? Explain.
A. [c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer: [d ◦ c](x) represents the cost of installation using the coupon and then the discount results in the lower cost.
B. [c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer: [c ◦ d](x) represents the cost of installation using the discount and then the coupon results in the lower cost.
C. [c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer: [c ◦ d](x) represents the cost of installation using the discount and then the coupon results in the lower cost.
D. [c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer: [c ◦ d](x) represents the cost of installation using the coupon and then the discount results in the lower cost.
Example 5