# Math Analysis – How to Derive the Inverse Function from a Rational Function

By | November 1, 2016

## Inverse Functions

In this tutorial students learn how to derive the inverse of a rational one-to-one function.  The students will have to understand a few key concepts to better grasp this topic.

First, rational functions have numerators and denominators that are both polynomials.

Second, an inverse function can only be created if the original function is one-to-one.  One-to-one means that each x value of the function has only 1 unique y value.  The vertical line test is a visual method to check to see if a function is one-to-one.

Third, for each point on the inverse function there exists a reversed order pair on original function.  Stated another way, the domain and range values are reversed for all points on the graph for both functions.

As an example the point (3,7) has an inverse at point (7,3).

Note that the x values are also called domain or input values.

Note that the y values are also called range or output values.

## Rational Function

$y=\frac {5x+2} {x-3}$

## Inverse Function

$y=\frac {3x+2} {x-5}$

## Graphing Tool

Click the link below open an interactive graphing calculator using the given and derived equations.  The line y=x is the line of symmetry.