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?

Technology guidance for today’s lesson:

Compare Inverse Linear Functions in Tables and Graphs: Casio ClassPad Casio fx-9750GIII

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 $f (x)$ represented with a table and an equation.

Use the two representations and the relationships you learned in the previous lesson to find an equation for $f^{- 1} (x)$ .
What relationship do you see between the equations of $f (x)$ and $f^{- 1} (x)$ ?

$f (x) = x - 5$

$x$	$f (x)$
$- 1$	$- 6$
$0$	$- 5$
$1$	$- 4$
$2$	$- 3$
$3$	$- 2$

2.

This time, you are given $f (x)$ represented by an equation and a graph. The line $y = x$ is shown as a dotted line on the graph to help you.

Use the two representations and the relationships you learned in the previous lesson to find an equation for $f^{- 1} (x)$ .
What relationship do you see between the equations of $f (x)$ and $f^{- 1} (x)$ ?

$f (x) = \frac{1}{2} (x - 3)$

3.

Here’s another one where $f (x)$ is given to you as an equation and a graph.

Find the equation of $f^{- 1} (x)$ .
What relationship do you see between the equations of $f (x)$ and $f^{- 1} (x)$ ?

$f (x) = 3 (x - 1) + 2$

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

If $f (a) = b$ , then $f^{- 1} (b) = a$ .

4.

$f (x) = \frac{2}{5} (x - 3) - 4$

5.

$f (x) = \frac{2 x + 1}{7}$

The graph and the equation of $f (x)$ are given. Find the equation of $f^{- 1} (x)$ and graph it with $f (x)$ .

6.

$f (x) = {(x + 3)}^{2} - 1$

$Domain : x \geq - 3$

Equation of $f^{- 1} (x)$ :

7.

$f (x) = 2 \sqrt{(x - 1)}$

Equation of $f^{- 1} (x)$ :

Ready for More?

$f (x) = {\begin{cases} x + 3, & 0 \leq x < 4 \\ 2 (x - 4) + 7, & 4 \leq x < 10 \end{cases}$

Find the graph and equation of $f^{- 1} (x)$ .

Graph of $f^{- 1} (x)$ :

Equation of $f^{- 1} (x)$ :

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, $f (x) = {\begin{cases} - \frac{1}{2} x + 2, & - 6 \leq x \leq - 2 \\ x + 5, & - 2 < x \leq 3 \\ - 2 (x - 3) + 8, & 3 < x \leq 6 \end{cases}$

3.

Solve for $C$ , $F = \frac{9}{5} C + 32$