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

Answer : $f\; o\; g(x)=x\; \text {and}\; g\; o\; f(x)=x$

Hint : If $f:A\rightarrow B$ and $g:B\rightarrow C$ be two given functions then the composite of f  and g denoted by $g\; o\; f$

$\therefore(g \circ f): A \rightarrow C(g \circ f)(x)=g\{f(x)\} \forall x \in A$

And dom $(g\; o\; f)=$ dom f

Given : Here given that $f(x)=e^{x} \; \text {and}\; g(x)=log_{e}x$

Here we have find out $f\; o\; g$ and $g\; o\; f$

Solution :

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

Here we have,

\begin{aligned} &f(x)=e^{x} \text { and } g(x)=\log _{e} x \\ &\begin{aligned} \therefore f \circ g(x) &=f\{g(x)\} \\ &=f\left\{\log _{e} x\right\} \\ &=\log _{e}\left(e^{x}\right) \\ &=x \\ \therefore f \circ g(x) &=x \end{aligned} \end{aligned}

Now, we find out $g\; o\; f$, we have ,

\begin{aligned} &f(x)=e^{x} \& g(x)=\log _{e} x \\ &\therefore g \circ f(x)=g\{f(x)\} \\ &\qquad \begin{aligned} &=g\left\{e^{x}\right\} \\ &=\log _{e}\left(e^{x}\right) \\ &=x \end{aligned} \\ &\therefore g \circ f=x \end{aligned}