Q. 72 - Precalculus Enhanced with Graphing Utilities

Question

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

$$\begin{gathered} f\left(x\right)=\frac{x^2+3}{3x^2} \\ x>0 \end{gathered}$$

Step-by-Step Solution

Answer

The inverse of the function$f\left(x\right)=\frac{x^2+3}{3x^2}$ is

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

Step 1. Given data

The given function is  

$f\left(x\right)=\frac{x^2+3}{3x^2}$

Step 2. interchanging variables

Replace with y and interchange x and y

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

Step 3. The inverse of the function

Solve the equation for y 

$$\begin{gathered} x=\frac{y^2+3}{3y^2} \\ 3x{y}^2=y^2+3 \\ 3x{y}^2-y^2=3 \\ {y}^2(3x-1)=3 \\ {y}^2=\frac{3}{3x-1} \\ y=\sqrt{\frac{3}{3x-1}} \end{gathered}$$

replace y with $f^{-1}\left(x\right)$

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

Step 4. Verification

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

$$\begin{gathered} f\left(f^{-1}\right(x\left)\right)=f\left(\sqrt{\frac{3}{3x-1}}\right) \\ f\left(f^{-1}\right(x\left)\right)=\frac{{\left(\sqrt{\frac{3}{3x-1}}\right)}^2+3}{3{\left(\sqrt{\frac{3}{3x-1}}\right)}^2} \\ f\left(f^{-1}\right(x\left)\right)=\frac{\frac{3}{3x-1}+3}{3\frac{3}{3x-1}} \\ f\left(f^{-1}\right(x\left)\right)=\frac{3+3(3x-1)}{9} \\ f\left(f^{-1}\right(x\left)\right)=\frac{3+9x-3}{9} \\ f\left(f^{-1}\right(x\left)\right)=\frac{9x}{9} \\ f\left(f^{-1}\right(x\left)\right)=x \end{gathered}$$

Determine  $$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=f^{-1}\left(\frac{x^2+3}{3x^2}\right) \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{3}{3\left(\frac{x^2+3}{3x^2}\right)-1}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{3x^2}{x^2+3-x^2}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{\frac{3x^2}{3}} \\ {f}^{-1}\left(f\right(x\left)\right)=\sqrt{x^2} \\ {f}^{-1}\left(f\right(x\left)\right)=x \end{gathered}$$

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