mac1105_lecture21_1

mac1105_lecture21_1 - L21 Inverse Functions Recall: The...

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231 L21 Inverse Functions Recall : The inverse of a function f is a function itself if and only if f is one-to-one. Let f be a one-to-one function. Then g is the inverse function of f if () ( ) fg x x = D for all x in the domain of g ; ( ) gfx x = D for all x in the domain of f . If g is the inverse function of f , then we write g as 1 f x and read: “ f -inverse”. Example : Determine whether the following functions are inverses of each other: (a) 5 2 fx x =− and 5 2 gx x = +
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232 (b) 2 () 1 fx x =+ , 0 x , and 1 gx x = −− .
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233 Cancellation Rules for Inverses The inverse functions undo each other with respect to the compositions: 1 (( ) ) ff x x = for all x in the domain of f 1 ) ) y y = for all y in the domain of 1 f Equivalent Form of the Cancellation Rules : 1 () f x yf y x =⇔ = ( x in domain of f ) ( y in the domain of 1 f ) Note on the Domains and Ranges of the Inverses : Domain of 1 f = Range of f Range of 1 f = Domain of f
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234 Graphing Inverses: If the graph of f is the set of points (, ) x y , then the graph of 1 f
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mac1105_lecture21_1 - L21 Inverse Functions Recall: The...

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