Chapter 4.3, Problem 8QR

To determine

To Find: The inverse of the given function.

Answer to Problem 8QR

The inverse is $f^{-1}(x)=\frac{8}{x}$

Explanation of Solution

Given information: Given function $y=\frac{8}{x}$

Calculation:

First we have to prove that the given function $f(x)=\frac{8}{x}$ is one-one onto(bijective).

For one-one, we have

  $\begin{array}{l}f(x_1)=f(x_2)\\ \frac{8}{x_1}=\frac{8}{x_2}\\ x_1=x_2\\ \therefore f\text{is}\text{one-one}\text{.}\end{array}$

For onto, we have

  $\begin{array}{l}y=\frac{8}{x}\\ x=\frac{8}{y}.\end{array}$

Now, if $y\in R-\left\{0\right\}$ , then there exists $x=\frac{8}{y}\in R$ such that

  $\begin{array}{l}f(x)=f\left(\frac{8}{y}\right)=y\\ \therefore f\text{is}\text{onto}\text{.}\end{array}$

Thus, $f$ is bijective and therefore invertible.

Now,

  $\begin{array}{l}f\circ f^{-1}(x)=x\\ f(f^{-1}(x))=x\\ \frac{8}{f^{-1}(x)}=x\\ \Rightarrow f^{-1}(x)=\frac{8}{x}\end{array}$