Question

the function f is one-to-one. Find its inverse and check your answer.

$$\begin{gathered} f\left(x\right)=\frac{x^2-4}{2x^2} \\ x>0 \end{gathered}$$

Short answer

The inverse of the function$f\left(x\right)=\frac{x^2-4}{2x^2}$ is$f^{-1}\left(x\right)=\sqrt{\frac{-4}{2x-1}},x>\frac{1}{2}$

Step-by-step solution

Step 1. Given data 

The given function is

$f\left(x\right)=\frac{x^2-4}{2x^2}$

Step 2. interchanging variables 

Replace with y and interchange x and y

$$\begin{gathered} f\left(x\right)=\frac{x^2-4}{2x^2} \\ y=\frac{x^2-4}{2x^2} \\ x=\frac{y^2-4}{2y^2} \end{gathered}$$

Step 3. The inverse of the function 

Solve the equation for y

$$\begin{gathered} x=\frac{y^2-4}{2y^2} \\ x\left(2y^2\right)=y^2-4 \\ 2x{y}^2=y^2-4 \\ 2x{y}^2-y^2=-4 \\ {y}^2(2x-1)=-4 \\ {y}^2=\frac{-4}{2x-1} \\ y=\sqrt{\frac{-4}{2x-1}} \end{gathered}$$

replace ywith $f^{-1}\left(x\right)$

role="math" localid="1646161961147" $f^{-1}\left(x\right)=\sqrt{\frac{-4}{2x-1}}$

Step 4. Verification 

Determine$$\begin{gathered} f\left(f^{-1}\right(x\left)\right)=f\left(\sqrt{\frac{-4}{2x-1}}\right) \\ f\left(f^{-1}\right(x\left)\right)=\frac{{\left(\sqrt{\frac{-4}{2x-1}}\right)}^2-4}{2{\left(\sqrt{\frac{-4}{2x-1}}\right)}^2} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-4-4(2x-1)}{-8} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-4-8x-4}{-8} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-8x}{-8} \\ f\left(f^{-1}\right(x\left)\right)=x \end{gathered}$$

Determine $f^{-1}\left(f\right(x\left)\right)$

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=f^{-1}\left(\frac{x^2-4}{2x^2}\right) \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{-4}{2\left(\frac{x^2-4}{2x^2}\right)-1}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{-4x^2}{x^2-4-x^2}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{-4x^2}{-4}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{x^2} \\ {f}^{-1}\left(f\right(x\left)\right)=x \end{gathered}$$

$f^{-1}\left(f\right(x\left)\right)=x\&f\left(f^{-1}\right(x\left)\right)=x$so inverse function is correct