# 5.1 Inverse Functions

**Topics:**Function, Injective function, Inverse function

**Pages:**2 (399 words)

**Published:**April 24, 2013

One-to-One Functions

* One-to-one function is where each x-value corresponds to one y-value, and each y-value corresponds to only one x-value *

* Horizontal line test – a function is one-to-one if every horizontal line intersects the graph of the function at most once * Examples: Determine whether the following functions are one-to-one

* In general, a function that is either increasing or decreasing on its entire domain, such as must be one-to-one * Tests to Determine Whether a Function is One-to-One

* Show that f(a) = f(b) implies a = b; then the function is one-to-one * Every y-value corresponds to no more than one x-value; to show that the function is not one-to-one, find at least two x-values that produce the same y-value * Use the horizontal line test

* If the function either increases or decreases on its entire domain, then it is one-to-one

Inverse Functions

* Let be a one-to-one function. Then g is the inverse function of if: *

* Examples: Determine whether f and g are inverses *

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* Inverse of a function is denoted as

* By definition of inverse function, the domain of f is the range of and the range of f is the domain of. * To find the inverse of a one-to-one function, interchange the x- and y-values of each of its ordered pairs. * Functions that are not one-to-one do not have an inverse

* Examples: If the functions are one-to-one, find their inverses. *

* Steps for Finding the Equation of the Inverse of a Function * Interchange x and y

* Solve for y

* Replace y with

* Examples: For each function that is one-to-one, a) write an equation for the inverse function in the form , b) graph the function & its inverse, c) give the domain and range of the function...

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