A series of free, online

Intermediate Algebra Lessons

or Algebra II lessons.

Videos, worksheets, and activities

to help Algebra students.

In these lessons, we will learn

- the definition of one-to-one functions
- the definition of inverse functions
- how to find the inverse of a function algebraically
- how to determine if two given functions are inverses
- how to find the inverse of a function graphically

### Definition of One-to-One Functions

After learning the definition of a function, we can extend it to define a one to one function. A one to one function has not only one output for every input, but also only one input in the domain for every output in the range. Another interesting type is an invertible function, or a function that has an inverse. The graph of a one to one or invertible function has unique and interesting characteristics.

The following diagram shows examples vertical and horizontal line tests. Scroll down the page for more examples and solutions on vertical and horizontal line tests for one-to-one functions.

A discussion on one-to-one functions and the horizontal line test.

One-to-one functions

### Definition of Inverse

Function

The definition of a function can be extended to define the definition of inverse of a function. Along with one to one functions, invertible functions are an important type of function. The definition of inverse says that a function’s inverse switches its domain and range. The definition of inverse helps students to understand the unique characteristics of the graphs of invertible functions.

Define an inverse function. Determine if a function as an inverse function. Determine inverse functions

All of the fundamental ideas of inverse functions are discussed. What exactly is an inverse function conceptually? Which functions have inverses? How do I graph inverse functions? What are the other basic relationships?

### Finding an Inverse Function Algebraically

Once we learn the definition of a function’s inverse we learn how to find the algebraic inverse, or how to find the inverse using algebraic methods. There are different methods for finding the inverse, the most common of which is to switch the dependent and independent variables and solve for the dependent variable. This is an important step in learning how to prove the inverse of a function.

Finding the Inverse of a Function.

This video outlines the procedure and do two complete examples of finding the inverse of a function.

Finding the Inverse of a Function or Showing One Does not Exist, Ex 2.

This video shows how to justify algebraically that a function does not have an inverse.

Finding the Inverse of a Function or Showing One Does not Exist, Ex 3.

This video shows how to find the inverse of a rational function.

Finding the Inverse of a Function or Showing One Does not Exist, Ex 4.

This video shows how to find the inverse of a root function.

### Proving Two Functions are Inverses

The definition of a function can be extended to define the definition of an inverse, or an invertible function. It’s important to understand proving inverse functions, and the method of proving inverse functions helps students to better understand how to find inverse functions. Students should review how to find an inverse algebraically and the basics of proofs.

This video provides two examples of determine if two given functions are inverses of one another by using

composition of functions

.

This video provides two examples of determine if two given functions are inverses of one another by using composition of functions.

Determine whether two functions are inverses.

Determine whether two functions are inverses.

### Finding an Inverse Function Graphically

In order to understand graphing inverse functions, students should review the definition of inverse functions, how to find the inverse algebraically and how to prove inverse functions. The graphs of inverse functions and invertible functions have unique characteristics that involve domain and range. Techniques for graphing inverse functions can make it easier to graph certain functions by hand.

How to find the inverse of a function given its graph

This video lesson covers the definition of the inverse of a relation and reviews the concept of find the inverse of a relation. Example 1 covers how to find the inverse of a relation from a table of values and provides a good visual of a relation that is a function and its inverse is not a function.

Before example 2, a discussion about how switching the x and the y in the equation is the best method for finding the inverse of a function. Example 2 models how to use the x and y switcheroo to find the inverse of a given function.

How to Find and Graph the Inverse of a Function

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