Section 1.4: Operations on Functions
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1 SECTION 14 Operations on Functions Section 14: 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: MATH 1330 Precalculus 107
2 CHAPTER 1 A Review of Functions Product: Quotient: Composition: Eample: 108 University of Houston Department of Mathematics
3 SECTION 14 Operations on Functions Eample: MATH 1330 Precalculus 109
4 CHAPTER 1 A Review of Functions Eample: 110 University of Houston Department of Mathematics
5 SECTION 14 Operations on Functions Additional Eample 1: MATH 1330 Precalculus 111
6 CHAPTER 1 A Review of Functions 11 University of Houston Department of Mathematics
7 SECTION 14 Operations on Functions Additional Eample : MATH 1330 Precalculus 113
8 CHAPTER 1 A Review of Functions Additional Eample 3: 114 University of Houston Department of Mathematics
9 SECTION 14 Operations on Functions Additional Eample 4: MATH 1330 Precalculus 115
10 CHAPTER 1 A Review of Functions 116 University of Houston Department of Mathematics
11 SECTION 14 Operations on Functions Additional Eample 5: MATH 1330 Precalculus 117
12 CHAPTER 1 A Review of Functions 118 University of Houston Department of Mathematics
13 Eercise Set 14: Operations on Functions Answer the following 1 f y g 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 f (a) Find f ( 3) g( 3) (b) Find f ( 0) g(0) (c) Find f ( 6) g( 6) (d) Find f ( 5) g(5) (e) Find f ( 7) g(7) (f) Sketch the graph of f g (Hint: For any value, add the y values of f and g) (g) What is the domain of f g? Eplain how you obtained your answer y g 3 f ( 3; g( f ( 5; g( f ( ; g( f ( ; g( f ( 6; g( 10 8 f ( 3; g( 4 9 f ( 9; g( 4 10 f ( 49 ; g( 3 Find the domain of each of the following functions 11 f ( 1 3 (a) Find f ( ) g( ) (b) Find f ( 0) g(0) (c) Find f ( 4) g( 4) (d) Find f ( ) g() (e) Find f ( 4) g(4) (f) Sketch the graph of f g (Hint: For any value, subtract the y values of f and g) (g) What is the domain of f g? Eplain how you obtained your answer h( 3 1 g ( f ( f ( 5 g ( 3 1 MATH 1330 Precalculus 119
14 Eercise Set 14: Operations on Functions Answer the following, using the graph below 17 (a) () (c) () 18 (a) (0) (c) (0) g (b) f g f (d) g f g (b) f g0 f (d) g f 0 19 (a) f g 3 (b) g f 3 0 (a) f g1 (b) g f 1 1 (a) f f 3 (b) g g (a) f f 5 (b) g g3 3 (a) f g4 (b) g f 4 4 (a) f g 5 (b) f g Use the functions f and g given below to evaluate the following epressions: f ( 3 and g( (a) (0) (c) (0) 6 (a) (1) (c) (1) f y g g (b) f g0 f (d) g f 0 g (b) f g1 f (d) g f 1 7 (a) f g (b) g f 8 (a) f g4 (b) g f 4 9 (a) f f 6 (b) g g6 30 (a) f f 4 (b) g g 4 31 (a) f g (b) g f 3 (a) f f (b) g g 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 standalone 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 f g If an -value is function of f g, ie 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 following problems: 1 33 f ( ; g( f ( ; g( 3 35 f ( ; g( f ( ; g( 3 For each of the following problems: (a) Find f g and its domain (b) Find g f and its domain 37 f ( 3; g( f ( 6 ; g( 7 f ( ; g( f ( ; g( 5 g for the 10 University of Houston Department of Mathematics
15 Eercise Set 14: Operations on Functions 41 f ( 7; g( 5 (a) (c) f gh1 (b) g h f 1 f g h (d) g h f 4 f ( 3 ; g( 9 Answer the following 43 Given the functions f ( and g ( 5 8, find: (a) (c) (e) (g) f g1 (b) g f 1 f g (d) g f f f 1 (f) g g1 f f (h) g g 44 Given the functions f ( 1 and g( 3, find: (a) f g3 (b) g f 3 (c) f g (d) g f (e) f f 3 (f) g g3 (g) f f (h) g g 1 45 Given the functions f ( and 3 g (, find: 5 (a) f g (b) g f (c) f g (d) g f 49 Given the functions f ( 4, g( 3, and h( 1, find: (a) (c) h f g4 (b) f gh0 f g h (d) h f g 50 Given the functions 1 f (, g(, and h( 3 4, find: (a) (c) h f g5 (b) f gh f g h (d) h f g Functions f and g are defined as shown in the table below f( ) g ( ) Use the information above to complete the following tables (Some answers may be undefined) f g 46 Given the functions f ( and 5 7 g (, find: 1 (a) (c) f g3 (b) g f 3 f g (d) g f 47 Given the functions f ( 1, g( 3 5, and h( 1, find: (a) (c) f gh (b) g h f 3 f g h (d) g h f g f f f g g 48 Given the functions f ( 3, g( 4, and h( 3, find: MATH 1330 Precalculus 11
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