Question

In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.

$f\left(x\right)=-\frac{2x}{x-1}$

Short answer

The inverse off is$\frac{x}{x+2}.$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=-\frac{2x}{x-1}$

We have to find its inverse and check the answer.

Step 2. Replace f(x) with y and interchange the variables x and y 

So,$y=-\frac{2x}{x-1}$

Interchange the variables

$x=-\frac{2y}{y-1}$

Step 3. Solve for y 

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

The inverse is$f^{-1}\left(x\right)=\frac{x}{x+2}.$

Step 4. Check the answer 

We can check the answers by $f^{-1}\left(f\left(x\right)\right)andf\left(f^{-1}\left(x\right)\right).$

So,

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=f^{-1}\left(-\frac{2x}{x-1}\right) \\ =\frac{\left(-{\displaystyle \frac{2x}{x-1}}\right)}{\left(-{\displaystyle \frac{2x}{x-1}}\right)+2} \\ =\frac{-2x}{-2} \\ =x \end{gathered}$$

Thus, $f^{-1}\left(f\left(x\right)\right)=x$

Now,

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=f\left(\frac{x}{x+2}\right) \\ =-\frac{2\left({\displaystyle \frac{x}{x+2}}\right)}{\left({\displaystyle \frac{x}{x+2}}\right)-1} \\ =\frac{-2x}{-2} \\ =x \end{gathered}$$

Thus, $f\left(f^{-1}\left(x\right)\right)=x$

Hence our result is right.