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mac1140_lecture18_1

# mac1140_lecture18_1 - L18 3.7 Operations and Compositions...

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154 L18 §3.7 Operations and Compositions Definition. Given two functions f and g . Let f D and g D be the domains of f and g . Then for all f g xD D ∈∩ , ( ) () fg x f x g x ±= ± ( ) ()() fg x f x g x = , and for all () 0 D g x , ff x x gg x ⎛⎞ = ⎜⎟ ⎝⎠ . Also, for all D f x , x x x =

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155 Example. Let () 5 fx x = and 2 4 gx x = . Find each of the following. Give the domain of each. ( ) fg x += f x g ⎛⎞ = ⎜⎟ ⎝⎠ g x f = ( 3 ) gf −=
156 The composition of f and g is () ( ) [ ( ) ] fg x f g x = D for all x in the domain of g such that ( ) gx is in the domain of f . We read the composition as “ f of g of x Similarly, ( ) [ ( ) ] gfx g f x = D for all x in the domain of f such that ( ) fx is in the domain of g .

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157 Example . For f and g given above, evaluate: a) ( )(9) gf = D b) ( )( ) fg x = D Domain: Important note : If () ( ) () Fx x = D then the domain of the composition D is the intersection of the domain of the “inner” function g and the resulting function F . That is, f gg F DD D = D
158 Example : Name two functions f and g such that: a) 32 4 () ( ) ( 1 ) fg x x x =+ + D b) 2 3 ( ) x x = D Hint : For this type of problems, first decide what you will consider an “inner” function ( ) gx . Then, rewrite the composition (() ) x only in terms of ( ) . Finally, give the “outer” function

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mac1140_lecture18_1 - L18 3.7 Operations and Compositions...

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