Q3: Where $f(x) = e^x$ is strictly increasing

To find where a function is strictly increasing, find where its derivative is positive. The exponential function is its own derivative:

$e^x$ is never 0 or negative. For $x \ge 0$⁠, $e^x \ge 1$⁠. For $x < 0$⁠, $e^x$ is 1 divided by $e^{-x}$⁠, and $e^{-x} > 1$⁠. So the graph lies wholly above the $x$-axis.

The rule: if $f'(x) > 0$ at every point of an interval, then $f$ is strictly increasing on that interval.

$f$ is increasing on $[1, \infty)$⁠, $(-\infty, 0)$ and $(0, \infty)$ too, but each is only a piece of where it increases. The interval asked for is the whole of $R$⁠.