Inverse Functions Rita Korsunsky

Definition x f(x) b f(b) a f(a) R D Definition

Horizontal Line Test If the graph of a function y = f(x) is such that no horizontal line intersects the graph in more than one point ,then it is one to one and has an inverse function. One-to-one function. Has an inverse function Not one-to-one. Has no inverse function

Definitions Example 1 f(x) and g(x) are inverse functions if : 1)  g(f(x)) = x for all x in the domain of f 2)  f(g(x)) = x for all x in the domain of g   Example 1

For every point (a,b) on the graph of f the point (b,a) is on the the graph of f-1 The graph of f-1 is the reflection of graph of f in line y= x

Example 2

Example 3

Guidelines for finding ƒ-1

Theorems

prove: Derivatives of Inverse functions (slopes of tangents) are reciprocals of each other at the points where domain x and range y are interchanged(scroll down to see slopes calculated)

Example

Example (when g(2) is not given)

Another proof(optional)