Chapter 11.6, Problem 23E

To determine

To find: the inverse of given function.

Answer to Problem 23E

The inverse of given function is $y=\pm \sqrt{\frac{3}{x-1}}$

Explanation of Solution

Given:

  $f\left(x\right)=\frac{3}{x^2}+1$

Calculation:

In this question, determine whether the inverse of $f\left(x\right)=\frac{3}{x^2}+1$ is a function and then find the inverse.

The graph of this function is

  

An infinite number of horizontal lines intersect the graph more than once. So the inverse of f is not a function.

Now, finding the inverse

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

Switch x and y.

  $f\left(x\right)=\frac{3}{x^2}+1$

Subtract 1 from both sides

  $x-1=\frac{3}{y^2}$

Cross multiply

  $y^2\left(x-1\right)=3$

Divide each side by $x-1$

  $y^2=\frac{3}{x-1}$

Take square root of both sides

  $y=\pm \sqrt{\frac{3}{x-1}}$

Conclusion:

The inverse of given function is $y=\pm \sqrt{\frac{3}{x-1}}$