Lesson 6 Well Versed Solidify 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 ?

A graph of y=x, which is dotted and a graph of a solid line passing through (-3, -3), (3, 0) and (6, 3) shown on the same coordinate plane x–5–5–5555101010y–5–5–5555101010000

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 ?

A graph of y=x, which is dotted and a graph of a solid line passing through (-1, -4), (0, -1) and (1, 2) shown on the same coordinate plane x–5–5–5555101010y–5–5–5555101010000

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

If , then .

4.

5.

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

6.

A graph of a curve beginning at (-3, -1), curving through (-2, 0) and (-1, 3) before exiting the grid just before (0, 8) x–5–5–5555y–5–5–5555000

Equation of :

a blank 17 by 17 grid

7.

A graph of a curve beginning at (1, 0), curving through (2, 2) and (5, 4) before exiting the grid just after (10, 6) x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000

Equation of :

a blank 17 by 17 grid

Ready for More?

Find the graph and equation of .

Graph of :

a blank 17 by 17 grid

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.

A continuous graph composed of four line segments beginning at closed endpoint (-6, 6) going to (-3, 3) then to (-1, 9), then to (3, 1), then to closed endpoint at (6, 4) x–5–5–5555y555101010000

2.

Graph the piecewise function,

a blank 17 by 17 grid

3.

Solve for ,