Question

In the following exercise, find the inverse of each function.

$f\left(x\right)=\sqrt[4]{9x-5}$

Short answer

The inverse of the given function is$f^{-1}\left(x\right)=\frac{x^4+5}{9}$

Step-by-step solution

Step 1. Given information 

The given function is$f\left(x\right)=\sqrt[4]{5x-9}$

Step 2.  Interchange the variable x and y.

Substituting $f\left(x\right)=y$, we get:

$y=\sqrt[4]{9x-5}$

Interchanging the variable $x$and $y$, we get:

$x=\sqrt[4]{9y-5}$

 Step 3. solve for  y 

Solving for $y$, we get:

$$\begin{gathered} x=\sqrt[4]{9y-5} \\ {x}^4=9y-5\left[Rai\sin gexponents4bothsides\right] \\ 9y=x^4+5 \\ y=\frac{x^4+5}{9} \end{gathered}$$

Substituting $y=f^{-1}\left(x\right)$, we get:

localid="1644914370168" $f^{-1}\left(x\right)=\frac{x^4+5}{9}$

Step 4. Verify that the functions are inverses.   

Substituting $f^{-1}\left(x\right)$in $f^{-1}\left(f\left(x\right)\right)=x$, we get:

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=x \\ {f}^{-1}\left(\sqrt[4]{5x-9}\right)=x \\ \frac{{\left(\sqrt[4]{9x-5}\right)}^4+5}{9}=x \\ \frac{9x-5+5}{9}=x \\ \frac{9x}{9}=x \\ x=x \end{gathered}$$

This is true.

Substituting $f\left(x\right)$in $f\left(f^{-1}\left(x\right)\right)=x$, we get:

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=x \\ f\left(\frac{x^4+5}{9}\right)=x \\ \sqrt[4]{9\frac{x^4+5}{9}}-5=x \\ \sqrt[4]{x^4+5}-5=x \\ x+5-5=x \\ x=x \end{gathered}$$

This is also true.

Thus, the obtained inverse is correct