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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:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 110
Example:
Solution:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 114
Solution:
Additional Example 3:
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