The graph of y F x h is not the same as the graph of y F x h The graph of y F x

The graph of y f x h is not the same as the graph of

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The graph of ( ) y F x h = + is not the same as the graph of ( ) . y F x h = + The graph of ( ) y F x h = + represents a vertical shift of the graph of ( ) . y F x = 67. The graph of ( ) 2 6 F x x = + is translated 6 units up from the graph of ( ) 2 . f x x = 68. ( ) 6 G x x = + 69. The graph of ( ) 6 G x x = + is translated 6 units up from the graph of ( ) . g x x = 70. The graph of ( ) ( ) 2 6 F x x = is translated 6 units to the right from the graph of ( ) 2 . f x x = 71. ( ) 6 G x x = 72. The graph of ( ) 6 G x x = is translated 6 units to the right from the graph of ( ) . g x x = 73. (a) Choose any value x . Find the corresponding value of y on both graphs and compare these values. For example, choose 2. x = From the graph, ( ) ( ) 2 1 and 2 3. f g = = For any value x , the y -value for ( ) g x is 2 greater than the y -value for ( ) , f x so the graph of ( ) g x is a vertical translation of the graph of ( ) f x up 2 units. Therefore ( ) ( ) 2, g x f x = + that is 2. c = (b) Choose any value y . Find the corresponding value of x on both graphs and compare these values. For example, choose 3. y = From the graph, ( ) ( ) 6 3 and 2 3. f g = = For any value y , the x -value for ( ) g x is 4 less than the x -value for ( ) , f x so the graph of ( ) g x is a horizontal translation of the graph of ( ) f x to the left 4 units. Therefore ( ) ( ) 4 , g x f x = + that is 4. c =
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Section 2.7: Function Operations and Composition 273 Section 2.7: Function Operations and Composition In Exercises 1–8, 2 ( ) 5 2 f x x x = and g ( x ) = 6 x + 4. 1. [ ] ( ) ( ) 2 ( )(3) (3) (3) 5(3) 2(3) 6(3) 4 45 6 18 4 39 22 61 f g f g + = + = + + = + + = + = 2. ( ) ( ) 2 ( )( 5) ( 5) ( 5) [5( 5) 2( 5)] [6( 5) 4] 125 10 30 4 135 ( 26) 161 f g f g = = + = + − − + = − − = 3. ( ) ( ) 2 ( )(4) (4) (4) [5(4) 2(4)] [6(4) 4] [5(16) 2(4)] [24 4] 80 8 28 72(28) 2016 fg f g = = + = + = = = 4. ( ) ( ) 2 ( )( 3) ( 3) ( 3) [5( 3) 2( 3)] [6( 3) 4] [5(9) 2( 3)] [ 18 4] 45 6 14 51( 14) 714 fg f g = = + = ⋅ − + = + ⋅ − = = − 5. 2 ( 1) 5( 1) 2( 1) ( 1) ( 1) 6( 1) 4 5 2 7 6 4 2 f f g g = = + + = = − + 6. 2 (4) 5(4) 2(4) (4) (4) 6(4) 4 80 8 72 18 24 4 28 7 f f g g = = + = = = + 7. 2 2 2 ( )( ) ( ) ( ) (5 2 ) (6 4) 5 2 6 4 5 8 4 f g m f m g m m m m m m m m m = = + = = 8. 2 2 2 2 ( )(2 ) (2 ) (2 ) [5(2 ) 2(2 )] [6(2 ) 4] [5(4) 2(2 )] [12 4] (20 4 ) (12 4) 20 8 4 f g k f k g k k k k k k k k k k k k + = + = + + = + + = + + = + + 9. ( ) 3 4, ( ) 2 5 f x x g x x = + = i) ( )( ) ( ) ( ) (3 4) (2 5) 5 1 f g x f x g x x x x + = + = + + = ii) ( )( ) ( ) ( ) (3 4) (2 5) 9 f g x f x g x x x x = = + = + iii) 2 2 ( )( ) ( ) ( ) (3 4)(2 5) 6 15 8 20 6 7 20 fg x f x g x x x x x x x x = = + = + = iv) ( ) 3 4 ( ) ( ) 2 5 f f x x x g g x x + = = The domains of both f and g are the set of all real numbers, so the domains of f + g , f g , and fg are all ( ) , . −∞ ∞ The domain of f g is the set of all real numbers for which ( ) 0. g x This is the set of all real numbers except 5 2 , which is written in interval notation as ( ) ( ) 5 5 2 2 , , . −∞ 10. f ( x ) = 6 – 3 x , g ( x ) = –4 x + 1 i ) ( )( ) ( ) ( ) (6 3 ) ( 4 1) 7 7 f g x f x g x x x x + = + = + − + = − + ii) ( )( ) ( ) ( ) (6 3 ) ( 4 1) 5 f g x f x g x x x x = = − − + = + iii) 2 2 ( )( ) ( ) ( ) (6 3 )( 4 1) 24 6 12 3 12 27 6 fg x f x g x x x x x x x x = = + = − + + = + iv) ( ) 6 3 ( ) ( ) 4 1 f f x x x g g x x = = + The domains of both f and g are the set of all real numbers, so the domains of f + g , f g , and fg are all ( ) , . −∞ ∞
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