Inverse & Piecewise Function

Question

I'm only unsure about question five. Thank you.

My Work

You have a summer job that pays time and a half for overtime. That is, if you work more than 40 hours a week, your hourly wage for the extra hours is 1.5 times your normal hourly wage of $7.

1. Copy and complete table, showing your work as well as the answer.

2. Write a piecewise function that gives your weekly pay "P" in terms of the number "H" of hours you work.

3. Graph your function.

4. Use your function in question 2 above to find P(45). Interpret your answer by using units in your explanation.

5. Use your function in question 2 above to find P^-1(70). Interpret your answer by using units in your explanation.

Answer 1

So the function $P$ that you have created takes the number of hours you work as an input and returns the amount of money you make for working that many hours. Therefore, $P^{-1}$ will be the inverse function, whose input is an amount of money and whose output will be the number of hours that need to be worked to make that much money.

Given that, can you calculste $P^{-1}(70)$?

Answer 2

You found the table of values of $h$ and $P(h)=\begin{cases}\quad \quad 7h \quad \quad,\ \ 0\le h\le 40\\ 10.5h-140,40<h\end{cases}$: $$\begin{array}{cc} h&P(h)\\ \hline 0&0\\ 2&14\\ 5&35\\ 40&220\\ 41&290.5\\ 60&490\\ \color{red}{?}&\color{red}{70}\end{array}$$ Can you find the number $h=\color{red}{?}$, for which $P(h)=\color{red}{70}$? This is an opposite (inverse) question.

Answer:

$7h=70 \Rightarrow h=10.$ So, $h=P^{-1}(70)=10$.