Answer
$f(x) = -e^x$
Work Step by Step
The exponential function has the form:
$\dfrac{f(x+1)}{f(x)} = a(1)$
We will use the points $(0, -1)$ and $(1, -e)$ in equation (1) to obtain: $\dfrac{-e}{-1}=a \implies a=e$
Thus, we can write the equation of the function as:
$f(x) = C \cdot e^x$.
In order to compute the value of $C$, we will apply the point $x=0; y=f(x)=-1$ as follows:
$-1 = C \times e^{(0)} \\C= \dfrac{-1}{1} \\ C=-1$
Therefore, the required exponential function is of the form:
$f(x) = -e^x$
