Let $f:R\{-<!--\; -\; -->\frac{1}{2}\}\to <!--\; \to \; -->R$ and $g:R-<!--\; -\; -->\{-<!--\; -\; -->\frac{5}{2}\}\to <!--\; \to \; -->R$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$ then the domain of the function f(g(x)) is

The correct answer is:
$R-\{-\frac{5}{2}\}$

To find the domain of the composite function $f(g(x))$, we must satisfy two conditions:
1. $x$ must be in the domain of $g(x)$.
2. The output value of $g(x)$ must be within the domain of $f(x)$.

Let us identify the domain of each individual function:
For the function $g(x)=\frac{|x|+1}{2x+5}$, the denominator must not be zero:
$2x+5=0\Rightarrow x=-\frac{5}{2}$
Thus, the domain of $g(x)$ is $D_g=R-\{-\frac{5}{2}\}$.

For the function $f(x)=\frac{2x+3}{2x+1}$, the denominator must not be zero:
$2x+1=0\Rightarrow x=-\frac{1}{2}$
Thus, the domain of $f(x)$ is $D_f=R-\{-\frac{1}{2}\}$.

For the composite function $f(g(x))$ to be defined, we must have $g(x)\ne -\frac{1}{2}$. Let us check if there are any real solutions to the equation:
$g(x)=-\frac{1}{2}$
Substituting the definition of $g(x)$:
$\frac{|x|+1}{2x+5}=-\frac{1}{2}$
Cross-multiplying, we obtain:
$2(|x|+1)=-(2x+5)$
$2|x|+2=-2x-5$
Rearranging the terms yields:
$2|x|+2x=-7$

Let us analyze the left-hand side $2|x|+2x$:
- If $x\ge 0$, then $|x|=x$, which gives $2x+2x=4x\ge 0$.
- If $x<0$, then $|x|=-x$, which gives $2(-x)+2x=0$.
Since $2|x|+2x\ge 0$ for all real numbers $x$, and the right-hand side is $-7$, the equation has no real solutions.

This means that $g(x)$ is never equal to $-\frac{1}{2}$ for any real value $x$ in the domain of $g$.
Therefore, the domain of the composite function $f(g(x))$ is exactly the domain of $g(x)$, which is:
$R-\{-\frac{5}{2}\}$