Write the range of the function f(x) =

Question:

Write the range of the function f(x) = ex[x]x ∈ R.

Solution:

$f(x)=e^{x-[x]}, x \in \mathrm{R}$

We know that $x-[x]=\{x\}$, which is the fractional part of any number $x$.

Thus, $f(x)=e^{\{x\}}$

Also, $0 \leq\{x\}<1$

$\Rightarrow e^{0} \leq e^{\{x\}}$\Rightarrow 1 \leq f(x)

Thus range of $f(x)$ is $[1, e)$.