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

Answers (1)

Answer : $g\; o\; f \left ( x \right )=4x^{2}+12x+14 \; \text {and}\; f\; o\; g\; =2x^{2}+13$

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

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

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

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

$f(x)=2x+3,g(x)=x^{2}+5$

Solution :

Now first we find $g\; o\; f$

$g\; o\; f (x)= g\left ( 2x+3 \right )$

$=\left ( 2x+3\right )^{2}+5$

$g\; o\; f (x)= 4x^{2}+12x+9+5$ $\because \left [ \left ( a+b \right )^{2}=a^{2}+b^{2}+2ab \right ]$

$g\; o\; f (x)= 4x^{2}+12x+14$

Now we find $f\; o\; g$

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

$f\left ( x^{2}+5 \right )=2\left ( x^{2}+5 \right )+3$

$f\; o\; g(x)=2x^{2}+13$

Hence, $f\; o\; g(x)=2x^{2}+13\; \text {and}\; g\; o\; f(x)=4x^{2}+12x+14$

Then we find $f\; o\; g$

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

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

$f\; o\; g\left ( x \right )= 2x^{3}+x^{6}$

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

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