Chapter 7.2, Problem 49PPS

(a)

To determine

To Find:

The inverse $F^{-1}(x)\text{ and show that F(x) and }{F}^{-1}(x)\text{ are inverses}\text{.}$

Answer to Problem 49PPS

  ${\text{F}}^{-1}(x)=\frac{5}{9}(x-32)$

Explanation of Solution

Given:

A formula for converting degrees Celsius to Fahrenheit is $F(x)=\frac{9}{5}x+32$

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:

  $\begin{array}{l}Given\\ F(x)=\frac{9}{5}x+32\\ \text{Rewrite it by replacing F(x) by 'y':}\\ y=\frac{9}{5}x+32\\ \text{Exchange x and y:}\\ x=\frac{9}{5}y+32\\ \text{Solve for y :}\\ x-32=\frac{9}{5}y\mathrm{..........}[subtract\text{ 32 from each side]}\\ y=(x-32)\frac{5}{9}\mathrm{..........}[multiply\text{ each side by }\frac{5}{9}\text{]}\\ {\text{So, F}}^{-1}(x)=\frac{5}{9}(x-32)\\ \\ \\ \text{f and g are inverse functions if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\\ \\ \text{We have }\\ F(x)=\frac{9}{5}x+32\\ {\text{F}}^{-1}(x)=\frac{5}{9}(x-32)\\ \left[F\circ F^{-1}\right](x)=F\left[F^{-1}(x)\right]\\ =F\left(\frac{5}{9}(x-32)\right)\\ =\frac{9}{5}\left(\frac{5}{9}(x-32)\right)+32\\ =x-32+32\\ =x\\ \left[F^{-1}\circ F\right](x)=F^{-1}\left[F(x)\right]\\ =F^{-1}\left(\frac{9}{5}x+32\right)\\ =\frac{5}{9}\left(\frac{9}{5}x+32-32\right)\\ =x\\ Therefore,\\ {\text{F(x) and F}}^{-1}(x)\text{ are inverses}\text{.}\end{array}$

(b)

To determine

To Explain:

The purpose of ${\text{F}}^{-1}(x)$

Answer to Problem 49PPS

  ${\text{F}}^{-1}(x)\text{ converts degree Fahrenheit to Celsius}\text{.}$

Explanation of Solution

Calculation:

  $\begin{array}{l}\text{Since F(x) is a function that converts the degree }\\ \text{Celsius to Fahrenheit}\text{.}\\ {\text{So, F}}^{-1}(x)\text{ , being the inverse of F(x) is a function }\\ \text{that converts degree Fahrenheit to Celsius}\text{.}\end{array}$