Chapter 1.9, Problem 89E

(a)

To determine

To find: the inverse function, and to find given variable represent in the inverse function.

Answer to Problem 89E

  $y=\frac{4x-40}{3}$ represents the inverse function of $f\left(x\right)$

Explanation of Solution

Given information:

Your wage is $10.00 per hour plus $0.75 for given unit produced per hour. So, your hourly wage $y$ in terms of the number of units produced $x$ is

  $y=10+0.75x$ .

Calculation:

We are given hourly wage $y$ in terms of the number of units produced $x$ as

  $y=10+0.75x$

Let $f\left(x\right)=10+0.75x$

Let us find the inverse $f\left(x\right)$ algebraically by first writing down the original functions which is $f\left(x\right)=10+0.75x$ .

We will now replace $f\left(x\right)$ by $y$ and interchange the $x$ and $y$ ,

  $\begin{array}{l}y=10+0.75x\\ x=10+0.75y\end{array}$

Finally solve for $y$ and replace by $f^{-1}\left(x\right)$

  $\begin{array}{c}y=\frac{x-10}{0.75}\\ =\frac{4x-40}{3}\\ y=\frac{4x-40}{3}\\ f^{-1}\left(x\right)=\frac{4x-40}{3}\end{array}$

  $y=\frac{4x-40}{3}$ represents the inverse function of $f\left(x\right)$

  $x$ represents hourly wage and

  $y$ represents number of units produced.

(b)

To determine

To find: the number of units produced when your hourly wage is $24.25.

Answer to Problem 89E

19 units

Explanation of Solution

Calculation:

To find the number of units produced when hourly wage is $24.25,

Let us consider the following

  $y=\frac{4x-40}{3}$

Putting the value of $x=24.25$ ,

  $\begin{array}{c}y=\frac{4\left(24.25\right)-40}{3}\\ =\frac{97-40}{3}\\ =\frac{57}{3}\\ =19\text{units}\end{array}$