Prove that

Question:
Let $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$. Find $g$ of and $f \circ g$.

Solution:

To find: $g$ o $f$ and $f \circ g$

Formula used: (i) $f \circ g=f(g(x))$

(ii) $g \circ f=g(f(x))$

Given: (i) $f(x)=8 x^{3}$

(ii) $g(x)=x^{1 / 3}$

We have,

$g \circ f=g(f(x))=g\left(8 x^{3}\right)$

$g \circ f=\left(8 x^{3}\right)^{\frac{1}{3}}=2 x$

$f \circ g=f(g(x))=f\left(x^{1 / 3}\right)$

$f \circ g=8\left(x^{\frac{1}{3}}\right)^{3}=8 x$

Ans) $g \circ f=2 x$ and $f \circ g=8 x$