# Lesson 16

Finding and Interpreting Inverse Functions

• Let’s find the inverse of linear functions.

Tickets to a family concert cost $10 for adults and$3 for children. The concert organizers collected a total of $900 from ticket sales. 1. In this situation, what is the meaning of each variable in the equation $$10A + 3C = 900$$? 2. If 42 adults were at the concert, how many children attended? 3. If 140 children were at the concert , how many adults attended? 4. Write an equation to represent $$C$$ as a function of $$A$$. Explain what this function tells us about the situation. 5. Write an equation to represent $$A$$ as a function of $$C$$. Explain what this function tell us about the situation. ### Problem 2 A school group has$600 to spend on T-shirts. The group is buying from a store that gives them a $5 discount off the regular price per shirt. $$n=\dfrac{600}{p-5}$$ gives the number of shirts, $$n$$, that can be purchased at a regular price, $$p$$. $$p=\dfrac{600}{n}+5$$ gives the regular price, $$p$$, of a shirt when $$n$$ shirts are bought. 1. What is $$n$$ when $$p$$ is 20? 2. What is $$p$$ when $$n$$ is 40? 3. Is one function an inverse of the other? Explain how you know. ### Problem 3 Functions $$f$$ and $$g$$ are inverses, and $$f(\text-2)=3$$. Is the point $$(3,\text-2)$$ on the graph of $$f$$, on the graph of $$g$$, or neither? ### Problem 4 Here are two equations that relate two quantities, $$p$$ and $$Q$$: $$Q=7p + 1,\!999$$ $$p=\dfrac{Q-1,999}{7}$$ Select all statements that are true about $$p$$ and $$Q$$. A: $$Q=7p + 1,\!999$$ could represent a function, but $$p=\dfrac{Q-1,999}{7}$$ could not. B: Each equation could represent a function. C: $$p=\dfrac{Q-1,999}{7}$$ could represent a function, but $$Q=7p + 1,\!999$$ could not. D: The two equations represent two functions that are inverses of one another. E: If $$Q=7p + 1,\!999$$ represents a function, then the inverse function can be defined by $$p=7Q-1,\!999$$. ### Problem 5 Elena plays the piano for 30 minutes each practice day. The total number of minutes $$p$$ that Elena practiced last week is a function of $$n$$, the number of practice days. Find the domain and range for this function. (From Unit 4, Lesson 10.) ### Problem 6 The graph shows the attendance at a sports game as a function of time in minutes. 1. Describe how attendance changed over time. 2. Describe the domain. 3. Describe the range. (From Unit 4, Lesson 11.) ### Problem 7 Two children set up a lemonade stand in their front yard. They charge$1 for every cup. They sell a total of 15 cups of lemonade. The amount of money the children earned, $$R$$ dollars, is a function of the number of cups of lemonade they sold, $$n$$.

1. Is 20 part of the domain of this function? Explain your reasoning.
2. What does the range of this function represent?
3. Describe the set of values in the range of $$R$$.
4. Is the graph of this function discrete or continuous? Explain your reasoning.
(From Unit 4, Lesson 11.)

### Problem 8

Here is the graph of function $$f$$, which represents Andre's distance from his bicycle as he walked in a park.

1. Estimate $$f(5)$$.
2. Estimate $$f(17)$$.
3. For what values of $$t$$ does $$f(t)=8$$?
4. For what values of $$t$$ does $$f(t)=6.5$$?
5. For what values of $$t$$ does $$f(t)=10$$?
(From Unit 4, Lesson 6.)