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ca_m6_handouts - 1 Rev.S08 MAC 1105 Module 6 Composite...

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Unformatted text preview: 1 Rev.S08 MAC 1105 Module 6 Composite Functions and Inverse Functions 2 Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. Perform arithmetic operations on functions. 2. Perform composition of functions. 3. Calculate inverse operations. 4. Identity one-to-one functions. 5. Use horizontal line test to determine if a graph represents a one-to-one function. 6. Find inverse functions symbolically. 7. Use other representations to find inverse functions. ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 Rev.S08 Composite Functions and Inverse Functions ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules.- Combining Functions; Composite Functions Combining Functions; Composite Functions- Inverse Functions Inverse Functions There are two major topics in this module: 2 4 Rev.S08 Five Ways of Combining Functions If If f ( x ) and ) and g ( x ) both exist, the sum, difference, ) both exist, the sum, difference, product, quotient and composition of two functions product, quotient and composition of two functions f and and g are defined by are defined by ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules. ( ) ( ) ( ) ( ) )) ( ( ) ( ) ( where ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x g f x g f x g x g x f x g f x g x f x fg x g x f x g f x g x f x g f = ! = " # $ % & ’ ( = ) = ) + = + o 5 Rev.S08 Example of Addition of Functions ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let Let f ( x ) = ) = x 2 2 + 2 + 2 x and and g ( x ) = 3 ) = 3 x- 1- 1 Find the symbolic representation for the function Find the symbolic representation for the function f f + + g and use this to evaluate ( and use this to evaluate ( f f + + g )(2). )(2). ( f + g )( x ) = ( x 2 2 + 2 + 2 x ) + ( 3 x − 1) 1) ( f + g )( x ) = x 2 2 + 5 + 5 x − 1 1 ( f + g )(2) = 2 2 2 + 5( + 5(2) − 1 = 13 1 = 13 or or ( ( f f + + g )(2) = )(2) = f (2) + (2) + g (2) (2) = 2 = 2 2 2 + 2(2) + 3(2) + 2(2) + 3(2) − 1 1 = 13 = 13 6 Rev.S08 Example of Subtraction of Functions ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let Let f ( x ) = ) = x 2 2 + 2 + 2 x and and g ( x ) = 3 ) = 3 x − 1 1 Find the symbolic representation for the function f − g and use this to evaluate ( f − g )(2). ( f − g )( x ) = ( x 2 2 + 2 + 2 x ) − ( 3x 3x − 1) 1) ( f − g )( )( x ) = ) = x 2 2 − x + 1 + 1 So So ( f − g )(2) = 2 2 2 − 2 + 1 = 3 + 1 = 3 3 7 Rev.S08 Example of Multiplication of Functions ht p:/ faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let Let f ( x ) = ) = x 2 2 + 2 + 2 x and and g ( x ) = 3 ) = 3 x − 1 1 Find the symbolic representation for the function fg and use this to evaluate ( fg )(2) ( fg )( x ) = ( x 2 2 + 2 + 2 x ) ( 3 x − 1) 1) ( fg )( x ) = 3 x 3 3 + 6 + 6 x 2 − x 2 − 2 2 x ( fg )( x ) = 3 x 3 3 + 5 + 5 x 2 2 − 2 2 x So So (fg)(2) = 3(2) 3(2) 3 3 +5(2) +5(2) 2 2 − 2(2) = 40 2(2) = 40...
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