Question

Question- If $f\left( x \right) = 8{x^3}$, $g\left( x \right) = {x^{\dfrac{1}{3}}}$, then $fog\left( x \right)$ is
\[{\text{A}}{\text{. }}\left( {{8^3}} \right)x\]
${\text{B}}{\text{. }}{\left( {8x} \right)^{\dfrac{1}{3}}}$
${\text{C}}{\text{. }}8{x^3}$
${\text{D}}{\text{. }}8x$

Answer 1

Hint- Here, we will proceed by replacing $x$ with $g\left( x \right)$ in $f\left( x \right)$.

Given two functions $f\left( x \right) = 8{x^3}$ and $g\left( x \right) = {x^{\dfrac{1}{3}}}$
$fog\left( x \right) = f\left( {g\left( x \right)} \right) = f\left( {{x^{\dfrac{1}{3}}}} \right)$
The above function can be determined by replacing $x$ with ${x^{\dfrac{1}{3}}}$ in $f\left( x \right) = 8{x^3}$, we get
\[ \Rightarrow fog\left( x \right) = f\left( {g\left( x \right)} \right) = f\left( {{x^{\dfrac{1}{3}}}} \right) = 8{\left( {{x^{\dfrac{1}{3}}}} \right)^3} = 8x\]
Therefore, option D is correct.

Note- In these type of problems, in order to find the required function like $fog\left( x \right)$ we replace $x$ in $f\left( x \right)$ with $g\left( x \right)$ and similarly to find the function $gof\left( x \right)$ we replace $x$ in $g\left( x \right)$ with $f\left( x \right)$.