Question

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

$f\left(x\right)=-\frac{3x+1}{x}$

Short answer

The inverse off is$-\frac{1}{x+3}.$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=-\frac{3x+1}{x}$

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,$y=-\frac{3x+1}{x}$

Interchange the variables

$x=-\frac{3y+1}{y}.$

Step 3. Solve for y 

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

The inverse is$f^{-1}\left(x\right)=-\frac{1}{x+3}$

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,

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=f^{-1}\left(-\frac{3x+1}{x}\right) \\ =-\frac{1}{\left(-{\displaystyle \frac{3x+1}{x}}\right)+3} \\ =\frac{-x}{-1} \\ =x \end{gathered}$$

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

Now,

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=f\left(-\frac{1}{x+3}\right) \\ =-\frac{3\left(-{\displaystyle \frac{1}{x+3}}\right)+1}{\left(-{\displaystyle \frac{1}{x+3}}\right)} \\ =\frac{-x}{-1} \\ =x \end{gathered}$$

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

Hence our result is right.