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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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