# Lesson 15

Inverse Functions

Let’s define functions forward and backward.

### Problem 1

Noah's cousin is exactly 7 years younger than Noah. Let \(C\) represent Noah's cousin's age and \(N\) represent Noah's age. Ages are measured in years.

- Write a function that defines the cousin's age as a function of Noah's age. What are the input and output of this function?
- Write the inverse of the function you wrote. What are the input and output of this inverse function?

### Problem 2

Noah's cousin is exactly 7 years younger than Noah. Let \(M\) represent Noah's cousin's age in months and \(N\) represent Noah's age in years.

- If Noah is 15 years old, how old is his cousin, in months?
- When Noah's cousin is 132 months old, how old is Noah, in years?
- Write a function that gives the age of Noah's cousin in months, as a function of Noah's age in years.
- Write the inverse of the function you wrote. What are the input and the output of this inverse function?

### Problem 3

Each equation represents a function. For each, find the inverse function.

- $c=w+3$
- $y=x-2$
- $y=5x$
- $w=\frac{d}{7}$

### Problem 4

The number of years, \(y\), is a function of the number of months, \(m\). The number of months, \(m\), is also a function of the number of years, \(y\).

- Write two equations, one to represent each function.
- Explain why the two functions are inverses.

### Problem 5

Sketch a graph to represent each quantity described as a function of time. Be sure to label the vertical axis.

Swing: The height of your feet above ground while swinging on a swing at a playground

Slide: The height of your hat above ground as you walk to a slide, go up a ladder, and then go down a slide

Merry-go-round: Your distance from the center of a merry-go-round as you ride the merry-go-round

Merry-go-round, again: Your distance from your friend, who is standing next to the merry-go-round as you go around

### Problem 6

Lin charges \$5.50 per hour to babysit. The amount of money earned, in dollars, is a function of the number of hours that she babysits.

Which of the following inputs is impossible for this function?

-1

2

5

8

### Problem 7

The instructions for cooking a steak with a pressure cooker can be represented with this set of rules, where \(x\) represents the weight of a steak in ounces and \(f(x)\) the cooking time in minutes.

\(\displaystyle f(x)=\begin{cases} 7,& 8\leq x\leq 12 \\ 8, & 12< x\leq 13 \\ 9, & 13< x\leq 14\\ 10, & 14< x\leq 15\\ 11, & 15< x\leq 16\\ \end{cases} \)

- Describe the instructions in words so that they can be followed by someone using the pressure cooker.
- Graph function \(f\).

### Problem 8

The absolute value function \(Q(x)=|x|\) gives the distance from 0 of the point \(x\) on the number line.

\(Q\) can also be defined using piecewise notation: \(Q(x)=\begin{cases} x,& x\geq 0 \\ \text-x,& x < 0 \end{cases} \)

Determine if each point is on the graph of \(Q\). For each point that you believe is *not* on the graph of \(Q\), change the output coordinate so that the point is on the graph of \(Q\).

- \((\text- 3, 3)\)
- \((0,0)\)
- \((\text-5, \text-5)\)
- \((\text-72, 72)\)
- \((\frac45,\text- \frac45)\)