SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 107

Section 1.4: Operations on Functions

Combining Functions by Addition, Subtraction, Multiplication,

Division, and Composition

Combining Functions by Addition, Subtraction,

Multiplication, Division, and Composition

Definition of the Sum, Difference, Product, Quotient, and

Composition of Functions:

Sum:

Difference:

CHAPTER 1 A Review of Functions

University of Houston Department of Mathematics 108

Product:

Quotient:

Composition:

Example:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 109

Example:

Solution:

CHAPTER 1 A Review of Functions

University of Houston Department of Mathematics 110

Example:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 111

Additional Example 1:

Solution:

CHAPTER 1 A Review of Functions

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SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 113

Additional Example 2:

CHAPTER 1 A Review of Functions

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Solution:

Additional Example 3:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 115

Solution:

Additional Example 4:

CHAPTER 1 A Review of Functions

University of Houston Department of Mathematics 116

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 117

Additional Example 5:

Solution:

CHAPTER 1 A Review of Functions

University of Houston Department of Mathematics 118

Exercise Set 1.4: Operations on Functions

MATH 1330 Precalculus 119

x

y

f

g

x

y

f

g

Answer the following.

1.

(a) Find )3()3( gf .

(b) Find )0()0( gf .

(c) Find )6()6( gf .

(d) Find )5()5( gf .

(e) Find )7()7( gf .

(f) Sketch the graph of gf . (Hint: For any x

value, add the y values of f and g.)

(g) What is the domain of gf ? Explain how

you obtained your answer.

2.

(a) Find )2()2( gf .

(b) Find )0()0( gf .

(c) Find )4()4( gf .

(d) Find )2()2( gf .

(e) Find )4()4( gf .

(f) Sketch the graph of gf . (Hint: For any x

value, subtract the y values of f and g.)

(g) What is the domain of gf ? Explain how

you obtained your answer.

For each of the following problems:

(f) Find f g and its domain.

(g) Find f g and its domain.

(h) Find fg and its domain.

(i) Find f

g and its domain.

Note for (a)-(d): Do not sketch any graphs.

3. 124)(;32)( 2 xxxgxxf

4. 158)(;52)( 23 xxxgxxxf

5. 2

)(;1

3)(

x

xxg

xxf

6. 5

2)(;

5

4)(

x

xxg

xxf

7. xxgxxf 10)(;6)(

8. 4)(;32)( xxgxxf

9. 4)(;9)( 22 xxgxxf

10. 3)(;49)( 2 xxgxxf

Find the domain of each of the following functions.

11. 13

2)(

x

xxf

12. x

xxh3

2)(

13. 2

1

7

3)(

x

x

xxg

14. 71

5

6

2)(

xx

xxf

15. 5

2)(

x

xxf

16. 1

3)(

x

xxg

Exercise Set 1.4: Operations on Functions

University of Houston Department of Mathematics 120

Answer the following, using the graph below.

17. (a) )2(g (b) 2gf

(c) )2(f (d) 2fg

18. (a) )0(g (b) 0gf

(c) )0(f (d) 0fg

19. (a) 3gf (b) 3fg

20. (a) 1gf (b) 1fg

21. (a) 3ff (b) 2gg

22. (a) 5ff (b) 3gg

23. (a) 4gf (b) 4fg

24. (a) 5gf (b) 2gf

Use the functions f and g given below to evaluate the

following expressions:

45)( and 23)( 2 xxxgxxf

25. (a) )0(g (b) 0gf

(c) )0(f (d) 0fg

26. (a) )1(g (b) 1gf

(c) )1(f (d) 1fg

27. (a) 2gf (b) 2fg

28. (a) 4gf (b) 4fg

29. (a) 6ff (b) 6gg

30. (a) 4ff (b) 4gg

31. (a) xgf (b) xfg

32. (a) xff (b) xgg

The following method can be used to find the domain

of f g :

(a) Find the domain of g.

(b) Find f g .

(c) Look at the answer from part (b) as a stand-

alone function (ignoring the fact that it is a

composition of functions) and find its domain.

(d) Take the intersection of the domains found in

steps (a) and (c). This is the domain of f g .

Note:

We check the domain of g because it is the inner

function of f g , i.e. f g x . If an x-value is

not in the domain of g, then it also can not be an

input value for f g .

Use the above steps to find the domain of f g for the

following problems:

33. 2

1( ) ; ( ) 5f x g x x

x

34. 2

1( ) ; ( ) 2f x g x x

x

35. 2

3( ) ; ( ) 6

4f x g x x

x

36. 2

5( ) ; ( ) 3

2f x g x x

x

For each of the following problems:

(a) Find f g and its domain.

(b) Find g f and its domain.

37. 72)( ;3)( 2 xxgxxxf

38. 2( ) 6 2; ( ) 7f x x g x x

39. 4

1)( ;)( 2

xxgxxf

40. 2)( ;5

3)( xxg

xxf

x

y

f

g

Exercise Set 1.4: Operations on Functions

MATH 1330 Precalculus 121

41. xxgxxf 5)( ;7)(

42. xxgxxf 29)( ;3)(

Answer the following.

43. Given the functions 2)( 2 xxf and

85)( xxg , find:

(a) 1gf (b) 1fg

(c) xgf (d) xfg

(e) 1ff (f) 1gg

(g) xff (h) xgg

44. Given the functions 1)( xxf and

223)( xxxg , find:

(a) 3gf (b) 3fg

(c) xgf (d) xfg

(e) 3ff (f) 3gg

(g) xff (h) xgg

45. Given the functions 2

1)(

x

xxf and

5

3)(

xxg , find:

(a) 2gf (b) 2fg

(c) xgf (d) xfg

46. Given the functions 5

2)(

x

xxf and

1

7)(

x

xxg , find:

(a) 3gf (b) 3fg

(c) xgf (d) xfg

47. Given the functions

,21)( and,53)(,1)( 2 xxhxxgxxf find:

(a) 2hgf (b) 3fhg

(c) hgf (d) fhg

48. Given the functions

,23)( and,4)(,32)( 2 xxhxxgxxf find:

(a) 1hgf (b) 1fhg

(c) hgf (d) fhg

49. Given the functions

,12)( and,3)(,4)( 2 xxhxxgxxf find:

(a) 4gfh (b) 0hgf

(c) hgf (d) gfh

50. Given the functions

,43)( and,2)(,1

)(2

xxhxxgx

xf find:

(a) 5gfh (b) 2hgf

(c) hgf (d) gfh

Functions f and g are defined as shown in the table

below.

x 0 1 2 4

( )f x 2 4 4 7

( )g x 4 5 0 1

Use the information above to complete the following

tables. (Some answers may be undefined.)

51.

52.

53.

54.

x 0 1 2 4

f g x

x 0 1 2 4

g f x

x 0 1 2 4

f f x

x 0 1 2 4

g g x