#### Provide solution for RD Sharma Maths Class 12 Chapter Functions Exercise 2.2 Question 1 Sub question (vi).

Answer : $g\; o\; f=2x\; \text {and}\; f\; o\; g=8x$

Hint : $g\; o\; f$ means $f(x)$ function is in $g(x)$ function

$f\; o\; g$ means $g(x)$ function is in $f(x)$ function

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

$g(x)=x^{\frac{1}{3}}$

Solution :

Since  $f:R\rightarrow R \; \text {and} \; g:R\rightarrow R$

$f\; o\; g:R\rightarrow R \; \text {and} \; g\; o\; f:R\rightarrow R$

Now, $g\; o\; f(x)=g(f(x))=g(8x^{3} )$

$g\; o\; f(x)=(8x^{3} )^{\frac{1}{3}}$

$=\left ( 8 \right )^{\frac{1}{3}}=\left ( 2^{3}\right )^{\frac{1}{3}}$                                                                       $\because \left [ \text {as we know},\left ( a^{m} \right )^{\frac{1}{n}}=\left ( a \right )^{\frac{m}{n}} \right ]$

$g\; o\; f (x)=(8)^{\frac{1}{3}}(x^{3})^{\frac{1}{3}}$

$g\; o\; f (x)= 2x$

Similarly,

$f\; o\; g (x)=f(g(x))=f(x)^{\frac{1}{3}}$

$f\; o\; g (x)=8 (x^{\frac{1}{3}})^{3}$

$f\; o\; g (x)=8 x$                      $\because \left [ (a^{m})^{\frac{1}{n}}=(a)^{\frac{m}{n}} \right ]$

Hence, $f\; o\; g=8x\; \text {and}\; g\; o\; f=2x$