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Let $f(x)= \dfrac{1}{1+x^2}$ and $g(x)=\dfrac{e^{−x}}{1+[x]}$, where $[x]$ is the greatest integer less than or equal to $x$. Then which of the following domain is true?

1. domain of $(f+g)=R-(-2,-1]$
2. domain of $(f+g)=R-[-1,0)$
3. $[\text{range of f}] \cap [\text{range of g}]=\bigg[-2, \dfrac{1}{2} \bigg]$
4. $[\text{range of f}] \cap [\text{range of g}]= \bigg[- \dfrac{1}{2},\dfrac{1}{2} \bigg]-\{0\}$
1. Both II and IV
2. Both I and III
3. Both I and IV
4. Both II and III

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