Answer
$g(x)= \sqrt[3] {4x^2}$.
Work Step by Step
The given function is
$f(x)=\sqrt[3] {2x}$ and $(fg)(x)=2x$.
We can write $(fg)(x)=f(x)\cdot g(x)$.
Substitute all functions.
$\Rightarrow 2x=(\sqrt[3] {2x})\cdot g(x)$
Rewrite as an exponential expression.
$\Rightarrow 2x=(2x)^{\frac{1}{3}}\cdot g(x)$
Raise each factor in parentheses to the $\frac{1}{3}$ power.
$\Rightarrow 2x=(2^{\frac{1}{3}}x^{\frac{1}{3}})\cdot g(x)$
Divide both sides by $(2^{\frac{1}{3}}x^{\frac{1}{3}})$.
$\Rightarrow \frac{2x}{(2^{\frac{1}{3}}x^{\frac{1}{3}})}=\frac{(2^{\frac{1}{3}}x^{\frac{1}{3}})\cdot g(x)}{(2^{\frac{1}{3}}x^{\frac{1}{3}})}$
To divide with the same base, subtract exponents.
$\Rightarrow 2^{1-\frac{1}{3}}x^{1-\frac{1}{3}}= g(x)$
Simplify.
$\Rightarrow 2^{\frac{3}{3}-\frac{1}{3}}x^{\frac{3}{3}-\frac{1}{3}}= g(x)$
$\Rightarrow 2^{\frac{3-1}{3}}x^{\frac{3-1}{3}}= g(x)$
$\Rightarrow 2^{\frac{2}{3}}x^{\frac{2}{3}}= g(x)$
Rewrite in radical notation.
$\Rightarrow \sqrt[3] {2^2x^2}= g(x)$
$\Rightarrow \sqrt[3] {4x^2}= g(x)$.
