Chapter 6.2, Problem 40PPS

(a)

To determine

To Find:

The inverse of each function.

Answer to Problem 40PPS

  $M^{-1}(x)=\frac{x+22}{3}$

  ${\text{F}}^{-1}(x)=\frac{x+21}{3}$

Explanation of Solution

Given:

The shoe size for the average U.S. teen or adult male can be determined using the formula $M(x)=3x-22$ , where x is length of a foot in measured inches. The shoe size for the average U.S. teen or adult female can be found by using the formula $F(x)=3x-21$ .

Concept Used :

 A function  g is the inverse of a function  f if whenever

  $\begin{array}{l}y=f(x)\text{ then }x=g(y)\\ \text{That is ,}\\ \text{f and g are inverse functions if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\end{array}$

Calculation:

  $\begin{array}{l}Given\\ M(x)=3x-22\\ \text{Rewrite it by replacing M(x) by 'y':}\\ y=3x-22\\ \text{Exchange x and y:}\\ x=3y-22\\ \text{Solve for y :}\\ x+22=3y\mathrm{..........}[\text{add 22 each side]}\\ y=\frac{x+22}{3}\mathrm{..........}[\text{divide each side by 3]}\\ {\text{So, M}}^{-1}(x)=\frac{x+22}{3}\\ Given\\ F(x)=3x-21\\ \text{Rewrite it by replacing M(x) by 'y':}\\ y=3x-21\\ \text{Exchange x and y:}\\ x=3y-21\\ \text{Solve for y :}\\ x+21=3y\mathrm{..........}[\text{add 21 each side]}\\ y=\frac{x+21}{3}\mathrm{..........}[\text{divide each side by 3]}\\ {\text{So, F}}^{-1}(x)=\frac{x+21}{3}\end{array}$

(b)

To determine

To Find:

The size of Lucy’s feet if she wears size $7\frac{1}{2}$ shoe.

Answer to Problem 40PPS

  $9\frac{1}{2}\text{ inches}$

Explanation of Solution

Given:

From part (a) , we have :

  ${\text{F}}^{-1}(x)=\frac{x+21}{3}$

is a function where we get the size of female foot from the size of her shoe.

Calculation:

  $\begin{array}{l}Given\\ {\text{F}}^{-1}(x)=\frac{x+21}{3}\\ and\\ x=7\frac{1}{2}=\frac{15}{2}\\ \text{Substitute the value of x in the function:}\\ {\text{F}}^{-1}\left(7\frac{1}{2}\right)=\frac{x+21}{3}\\ \text{⇒}{\text{F}}^{-1}\left(7\frac{1}{2}\right)=\frac{\frac{15}{2}+21}{3}\\ \text{⇒}{\text{F}}^{-1}\left(7\frac{1}{2}\right)=9\frac{1}{2}\text{ inches}\\ \text{So, Lucy's feet is }9\frac{1}{2}\text{ inches long}\text{.}\end{array}$