Solve this
Question:

If $f(x)=\frac{1}{(2 x+1)}$ and $x \neq \frac{-1}{2}$ then prove that $f\{(x)\}=\frac{2 x+1}{2 x+3}$, when it is given

that $x \neq \frac{-3}{2}$

Solution:

Given: $f(x)=\frac{1}{(2 x+1)}$, where $x \neq \frac{-1}{2}$

Need to prove: $\mathrm{f}\{\mathrm{f}(\mathrm{x})\}=\frac{2 x+1}{2 x+3}$ when $x \neq \frac{-3}{2}$

Now placing f(x) in place of x

$\Rightarrow f\{(x)\}=\frac{1}{2 f(x)+1}$

$\Rightarrow f\{f(x)\}=\frac{1}{2 \frac{1}{2 x+1}+1}$

$f\{f(x)\}=\frac{1}{\frac{2+2 x+1}{2 x+1}}=\frac{2 x+1}{2 x+3}$, where $x \neq \frac{-3}{2}$ [Proved]