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Please solve RD Sharma class 12 Chapter Functions exercise 2.2 question 1 sub question (iii) maths textbook solution.

Answers (1)

Answer : $g\; o\; f = 3\left ( x^{2}+8 \right )^{3}+1\; \text {and}\; f\; o\; g=9x^{6}+6x^{3}+9$

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 : $f(x)=x^{2}+8$

$g(x)=3x^{3}+1$

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

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

Solution :

First, we find $g\; o\; f$

Thus, $g\; o\; f\left ( x \right )=g\left [ f\left ( x \right ) \right ]$

$g\; o\; f\left ( x \right )=g\left [ x^{2}+8 \right ]$

$g\; o\; f\left ( x \right )=3\left [ x^{2}+8 \right ]^{3}+1$

Similarly,

$f\; o\; g\left ( x \right )=f\left [ g(x) \right ]$

$f\; o\; g\left ( x \right )=f\left ( 3x^{3}+1 \right )$

$f\; o\; g\left ( x \right )= \left [ 3x^{3}+1 \right ]^{2}+8$

$f\; o\; g\left ( x \right )= \left [ 9x^{6}+1+6x^{3} \right ]^{2}+8$ $\because \left [ \left ( a+b \right )^{2} =a^{2}+b^{2}+2ab\right ]$

$f\; o\; g\left ( x \right )= 9x^{6}+1+6x^{3}+9$

Hence, $f\; o\; g = 9x^{6}+1+6x^{3}+9 \; \text {and}\; g\; o\; f=3\left ( x^{2}+8 \right )^{3}+1$

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