# Lesson 6Well VersedSolidify Understanding

## Learning Focus

Find the inverse of a function given any representation.

How can we find the equation of an inverse efficiently?

## Open Up the Math: Launch, Explore, Discuss

In this lesson, you will be using the characteristics of inverse functions to develop a deeper understanding of the relationship between the equations of inverse functions and a process for finding the equation of the inverse of a function. Keep an eye out for useful patterns and relationships between functions as you work the problems.

### 1.

You are given the function represented with a table and an equation.

1. Use the two representations and the relationships you learned in the previous lesson to find an equation for .

2. What relationship do you see between the equations of and ?

### 2.

This time, you are given represented by an equation and a graph. The line is shown as a dotted line on the graph to help you.

1. Use the two representations and the relationships you learned in the previous lesson to find an equation for .

2. What relationship do you see between the equations of and ?

### 3.

Here’s another one where is given to you as an equation and a graph.

1. Find the equation of .

2. What relationship do you see between the equations of and ?

Find the equation of the inverse function. Show that you have checked your work with the relationship:

If , then .

### 5.

The graph and the equation of are given. Find the equation of and graph it with .

Equation of :

### 7.

Equation of :

Find the graph and equation of .

Graph of :

Equation of :

## Takeaways

Finding inverse functions algebraically:

## Lesson Summary

In this lesson, we learned that the equation of an inverse function will contain the inverse operations in the reverse order. We used this idea to find a procedure to solve for the equation of the inverse of a function.

## Retrieval

### 1.

Create a piecewise function for the graph.

### 2.

Graph the piecewise function,

Solve for ,