Question

What is the inverse of the function g(x)=x^3/8+16?

Answer 1

I think this is the function

Answer 2

The inverse of the function for  $g^{}(x) = \frac{x^{3} }{8}+16$  is  $g^{-1}(x)=2\sqrt[3]{x-6}$

What is inverse of the function?

"The inverse function of a function f (also called the inverse of f ) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by  $f^{-1}$."

We have

$g^{}(x) = \frac{x^{3} }{8}+16$

Replace g(x) with y

$y =$ $\frac{x^{3} }{8}+16$

Interchange the variables

$x= \frac{y^{3} }{8}+16$

Rewrite the equation as

$\frac{y^{3} }{8}+16= x$

⇒$\frac{y^{3} }{8}= x-16$

Multiply both sides of the equation by 8

$8.\frac{y^{3} }{8}=8(x-16)$

⇒${y^{3} }=8(x-16)$

⇒${y^{3} }=8x-128$

⇒${y} }=\sqrt[3]{8x-128}$

⇒${y} }=2\sqrt[3]{x-6}$

Replace y with $g^{-1}(x)$

⇒$g^{-1}(x)=2\sqrt[3]{x-6}$

Hence, The inverse of the function for  $g^{}(x) = \frac{x^{3} }{8}+16$ is  $g^{-1}(x)=2\sqrt[3]{x-6}$

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