Q. 84 - Precalculus Enhanced with Graphing Utilities

Question

Find the inverse of the function $$\begin{gathered} f\left(x\right)=\sqrt{r^2-x^2} \\ 0\le x\le r \end{gathered}$$

Step-by-Step Solution

Answer

The inverse of the function $$\begin{gathered} f\left(x\right)=\sqrt{r^2-x^2} \\ 0\le x\le r \end{gathered}$$is  $$\begin{gathered} f^{-1}\left(x\right)=\sqrt{r^2-x^2} \\ 0\le x\le r \end{gathered}$$

Step 1. Given data

The given function is 

$$\begin{gathered} f\left(x\right)=\sqrt{r^2-x^2} \\ 0\le x\le r \end{gathered}$$

Step 2. interchanging variables

Replace $f\left(x\right)$ with y and interchange x and y 

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

Step 3. The inverse of the function

Solve the equation for y 

$$\begin{gathered} x=\sqrt{r^2-y^2} \\ {x}^2=r^2-y^2 \\ {y}^2=r^2-x^2 \\ y=\sqrt{r^2-x^2} \end{gathered}$$

replace y with $f^{-1}\left(x\right)$

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

Step 4. Verification

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

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

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

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

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