What properties of the exponential function causes $\mid e^{f(x)}| \leq e^{\mid f(x)\mid}$ to be true?

Question

I saw the fact that $\mid e^{f(x)}| \leq e^{\mid f(x)\mid}$ for all $f: \mathbb{R}\rightarrow \mathbb{R}$. Why must this be the case? I don't have an idea of what properties of the exponential function make this inequality true.

Answer 1

Since $g(x)=e^x$ increases and $f(x)\leq|f(x)|$, we obtain:

$$\mid e^{f(x)}|=e^{f(x)}\leq e^{\mid f(x)\mid}$$

Answer 2

For any $x \in \mathbb{R}$, we have $e^x \geq 0$, hence $|e^x| = e^x$. Also, $e^x$ is always increasing, and $|x| \geq x$. Therefore, $e^{|x|} \geq e^x = |e^x|$.