Find the domain and range of the following real function:
$f(x) = -|x|$

(N/A) The given function is $f(x) = -|x|$,where $x \in \mathbb{R}$.
We know that the absolute value function is defined as:
$|x| = \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0 \end{cases}$
Therefore,the function $f(x) = -|x|$ is defined as:
$f(x) = \begin{cases} -x, & \text{if } x \ge 0 \\ x, & \text{if } x < 0 \end{cases}$
Since $f(x)$ is defined for all real numbers $x$,the domain of $f$ is $\mathbb{R}$ (the set of all real numbers).
For any $x \in \mathbb{R}$,$|x| \ge 0$. Multiplying by $-1$,we get $-|x| \le 0$. Thus,$f(x) \le 0$ for all $x$.
Therefore,the range of $f$ is $(-\infty, 0]$.