Question

Let $f\left(x\right)=\frac{x}{1+x^2}$ and $g\left(x\right)=\frac{e^{-x}}{1+\left[x\right]}$, where $[.]$ represents the greatest integer function. Then the number of integral value(s) of $x$ which are not lying in the domain of $f+g$ is

Answer 1

$f\left(x\right)=\frac{x}{1+x^2}$
$f$ is defined for all real values of $x$.
Hence, $D\left(f\right)=R$

$g\left(x\right)=\frac{e^{-x}}{1+\left[x\right]}$
$g$ is not defined when $\left[x\right]=-1$
i.e., $x\in [-1,0)$
Hence, $D\left(g\right)=R-[-1,0)$

So, the domain of $f+g$ is $D\left(f\right)\cap D\left(g\right)$, which is $R-[-1,0)$
Hence, $-1$ does not lie in the domain of $f+g$