Chapter 11, Problem 10CR

To determine

To find: The inverse of the function $f^{-1}$ and its domain and range and, domain and range of $f$ .

Answer to Problem 10CR

The inverse of the function $f^{-1}=\frac{5-2x}{x}$ .

The domain and range of $f^{-1}$ are $R-\left\{0\right\}$ and $R$ .

The domain and range of $f$ are $R-\left\{-2\right\}$ and $R$ .

Explanation of Solution

Given information:

The function is

  $f\left(x\right)=\frac{5}{x+2}$

Calculation:

Obtain the inverse function.

  $\begin{array}{c}f\left(x\right)=\frac{5}{x+2}\\ f\left(x\right)\left(x+2\right)=5\\ xf\left(x\right)+2f\left(x\right)=5\\ x=\frac{5-2f\left(x\right)}{f\left(x\right)}\end{array}$

  $f^{-1}=\frac{5-2x}{x}$

The function $f\left(x\right)$ is defined for all values of $x$ except for $x=-2$ . The domain of the function is $R-\left\{-2\right\}$ , since the given function can take up any of the values in the range of real numbers the range of the given function is $R$ .

The function $f^{-1}$ is defined for all values of $x$ except for $x=0$ . The domain of the function is $R-\left\{0\right\}$ , since the given function can take up any of the values in the range of real numbers the range of the given function is $R$ .

Therefore, the inverse of the function $f^{-1}=\frac{5-2x}{x}$ , the domain and range of $f^{-1}$ are $R-\left\{0\right\}$ and $R$ , and the domain and range of $f$ are $R-\left\{-2\right\}$ and $R$ .