Answer
The domain of $g(t)$ is $(-\infty,\infty)$
The range of g(t) is $[-1,1]$
Work Step by Step
$$g(t)=\cos(e^{-t})=\cos\Big(\frac{1}{e^t}\Big)$$
1) Domain:
$g(t)$ is defined on $(-\infty,\infty)$ except where $e^t=0$
However, since $e^t\gt0$ for all $t\in R$, we know that $e^t\ne0$.
Therefore, the domain of $g(t)$ is $(-\infty,\infty)$
2) Range:
As stated above, $e^t\gt0$ for all $t\in(-\infty,\infty)$
So $\frac{1}{e^t}\gt0$ for all $t\in(-\infty,\infty)$
On the other hand, we know the range of a cosine function is $[-1, 1]$. And since the range of $\frac{1}{e^t}$ is $(0,\infty)$, in this range, $\cos\Big(\frac{1}{e^t}\Big)$ can have any values in its range $[-1,1]$
In other words, the range of g(t) is $[-1,1]$
