Answer
$ f(x)=-e^{x}$
Work Step by Step
The graph is all below the x-axis, falls, and passes through ($0,-1$)
$\Rightarrow $ The base is greater than 1, there has been a reflection across the x-axis.
We assume that $ f(x)=-b^{x}$
It passes through $(1,-e)$, so we assume that the base is $ e,\ f(x)=-e^{x}$, and we check with the other points:
$-(e)^{-1}=-\displaystyle \frac{1}{e}\quad $ ... point $(-1,\displaystyle \frac{1}{e})\quad $ ... OK
$-(e)^{2}=-e^{2}\quad $ ... point $(2,-e^{2})\quad $ ... OK
$ f(x)=-e^{x}$
