#### Please solve RD Sharma Class 12 Chapter Functions Exercise 2.3 Question 1 Sub question (ii) maths textbook solution.

\begin{aligned} &g \circ f(x)=\cos x^{2} \\ &f \circ g(x)=\cos ^{2} x \end{aligned}

Hint : Domain of f and domain g = R

Range of $f=(0,\infty )$

Range of g $g=(-1,1 )$

$\therefore$ Range of f $\subset$ domain of g

$\Rightarrow \; \; \; \; \; \; \; \; g\; o\; f$ exists

and Range of g $\subset$ domain of f

$\Rightarrow \; \; \; \; \; \; \; \; f\; o\; g$ exists

Given : Here given that

$f(x)=x^{2}\; \text {and}\; g(x)=\cos x$

Solution :

Here first we will find out $g\; o\; f,$

We have,

$\begin{array}{r} f(x)=x^{2} \text { and } g(x)=\cos x \\ \therefore g \circ f(x)=g\{f(x)\} \\ =g\left\{x^{2}\right\} \\ =\cos x^{2} \end{array}$

and again for $f\; o\; g$

\begin{aligned} &\therefore \operatorname{fog}(x)=f\{g(x)\} \\ &\qquad \begin{aligned} &=f\{\cos x\} \\ &=\cos ^{2} x \end{aligned} \end{aligned}

NOTE : $f\; o\; g$ is defined only when range $(g)\subseteq \text {dom}\; (f) \; \text {and dom}\;f\; o\; g=\text {dom}(g) .$