Chapter 5, Problem 18RE

To determine

To find : The inverse of the function and check the answer

Explanation of Solution

Given: The function is $f\left(x\right)=\frac{2-x}{3+x}$

The function can be written as,

  $y=\frac{2-x}{3+x}$

Interchange $x$ and $y$ terms.

  $\begin{array}{c}x=\frac{2-y}{3+y}\\ 3x+xy+y=2\\ \left(x+1\right)y=2-3x\\ y=\frac{2-3x}{x+1}\end{array}$

So, the inverse of the function is,

  $y=\frac{2-3x}{x+1}$

Check.

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=x\\ \frac{2-3\left(\frac{2-x}{3+x}\right)}{\left(\frac{2-x}{3+x}\right)+1}=x\\ \frac{5x}{5}=x\\ x=x\end{array}$

Hence, the result is correct.