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Unformatted text preview: 4.2 Inverse Functions Definition 4 (p.76) . For a given function f : E 1 → R 1 , E 1 ⊂ R 1 , we say a function g : E 2 → R 1 , E 2 ⊂ R 1 , is an inverse of f if g ( f ( x )) = x for all x ∈ E 1 and f ( g ( y )) = y for all y ∈ E 2 . • Example: f ( x ) = 3- 2 x , g ( y ) = 1 2 ( 3- y ) • If f has an inverse we say that f is invertible , and its inverse function is written as f- 1 . Ex Suppose f ( x ) = x 2 for x > 0. What is f- 1 ( x ) ? Figure 9: Graph of y = x 2 and its inverse function 4.3 The Derivative of the Inverse Function • The derivative of the inverse function f- 1 has a close relationship with the derivative of f . Theorem 8 (Thm 4.3; Inverse Function Theorem, p.79) . Let f be a C 1 function defined on the interval I ⊂ R 1 . If f prime ( x ) negationslash = for all x ∈ I, then (a) f is invertible on I, (b) its inverse g is C 1 function on the interval f ( I ) , and (c) for all y ∈ f ( I ) , g prime ( y ) = 1 f prime ( g ( y )) . 13 Proof. Since f is either increasing or decreasing on I (why?), f- 1 is well defined on f ( I ) . For each y ∈ f ( I ) , we have f ( g ( y ))= y . By differentiating both sides with respect to y , we obtain f prime ( g ( y )) · g prime ( y ) = 1. Hence g prime ( y ) = 1 f prime ( g ( y )) ,...
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This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.
- Spring '11