Question

Find the inverse of the function ,$f\left(x\right)=\sqrt[4]{6x-7}$

Short answer

The inverse of,$f\left(x\right)=\sqrt[4]{6x-7}$is

$f^{-1}\left(x\right)=\frac{x^4+7}{6}$

Step-by-step solution

Given information

Given the function$f\left(x\right)=\sqrt[4]{6x-7}$

Replace f(x) with y and then x,y

Replacing, we get

$$\begin{gathered} f\left(x\right)=\sqrt[4]{6x-7} \\ y=\sqrt[4]{6x-7} \\ x=\sqrt[4]{6y-7} \\ {x}^4=6y-7 \\ \frac{x^4+7}{6}=y \end{gathered}$$

Substitute f-1(x) for y

Substituting, we get

$f^{-1}\left(x\right)=\frac{x^4+7}{6}$

Verify f(f-1(x))=x

Substituting, we get

$$\begin{gathered} f\left(\frac{x^4+7}{6}\right)=\sqrt[4]{6\left(\frac{x^4+7}{6}\right)-7} \\ f\left(\frac{x^4+7}{6}\right)=\sqrt[4]{x^4+7-7} \\ f\left(\frac{x^4+7}{6}\right)=\sqrt[4]{x^4} \\ f\left(\frac{x^4+7}{6}\right)=x \end{gathered}$$

Verify f-1(f(x))=x

Substituting, we get

$$\begin{gathered} f^{-1}\left(\sqrt[4]{6x-7}\right)=\sqrt[4]{6\left(\frac{x^4+7}{6}\right)-7} \\ {f}^{-1}\left(\sqrt[4]{6x-7}\right)=\sqrt[4]{x^4+7-7} \\ {f}^{-1}\left(\sqrt[4]{6x-7}\right)=\sqrt[4]{x^4} \\ {f}^{-1}\left(\sqrt[4]{6x-7}\right)=x \end{gathered}$$

So these are inverses