Question

Determine range of the function $y=\frac{e^{-x}}{1+\left[x\right]}$
$\therefore$ Range $y=R-\left\{k\right\}$
what is k?

Answer 1

The function is defined for $1+\left[x\right]\ne 0,i.e.,x\text{/}ϵ(-1,0)$
$\because 1+\left[x\right]=0$ if $\left[x\right]=-1i.e.,xϵ(-1,0)$
$y=\frac{e^{-x}}{1+\left[x\right]}\Rightarrow (1+\left[x\right])y=e^{-x}$ Now $e^{-x}$ is always +ive.Hence above is possible if both the factors in L.H.S. are either+ive or both. -ive. $\therefore y>0$ and $1+\left[x\right]>0\Rightarrow xϵ(0,\infty )$ or $y<0$ and $1+\left[x\right]<0\Rightarrow xϵ(-\infty ,-1)$
$\therefore$ Range $y=R-\left\{0\right\}$