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:
$$R-\{-\frac{5}{2}\}$$

Step-by-Step Explanation:

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

Step 1: Determine the domain of $g(x)$
The function $g(x)$ is defined as:
$$g(x)=\frac{|x|+1}{2x+5}$$
For $g(x)$ to be real-valued and defined, its denominator must not be zero:
$$2x+5\ne 0\Rightarrow x\ne -\frac{5}{2}$$
Thus, the domain of $g(x)$ is:
$$D_g=R-\{-\frac{5}{2}\}$$

Step 2: Ensure $g(x)$ lies in the domain of $f(x)$
The domain of $f(x)$ is $R-\{-\frac{1}{2}\}$. Therefore, we must exclude any values of $x$ for which:
$$g(x)=-\frac{1}{2}$$
Let us solve the equation:
$$\frac{|x|+1}{2x+5}=-\frac{1}{2}$$
Cross-multiplying gives:
$$2(|x|+1)=-(2x+5)$$
$$2|x|+2=-2x-5$$
Rearranging the terms:
$$2|x|+2x=-7$$
$$2(|x|+x)=-7$$

For any real number $x$, the absolute value $|x|$ is always greater than or equal to $-x$ (i.e., $|x|\ge -x$).
This implies that:
$$|x|+x\ge 0$$
Consequently:
$$2(|x|+x)\ge 0$$
Since the left-hand side of our equation is always non-negative ( $\ge 0$) and the right-hand side is strictly negative ( $-7$), there are no real values of $x$ that satisfy this equation.
Therefore, $g(x)\ge 0$ for all $x>-\frac{5}{2}$, and $g(x)<0$ for $x<-\frac{5}{2}$, but $g(x)$ never equals $-\frac{1}{2}$ for any real number $x$.

Conclusion:
Since no points from the domain of $g(x)$ need to be excluded to satisfy the second condition, the domain of $f(g(x))$ is simply the domain of $g(x)$:
$$\text{Domain}=R-\{-\frac{5}{2}\}$$