Q.Find the domain and range of the modulus function $f(x) = |x|$, $x \in \mathbb{R}$, and sketch its graph.

The modulus function is defined for every real number and never produces a negative output.

The modulus (absolute value) function is defined piecewise as

Domain: the formula $|x|$ can be evaluated for every real number $x$ — there is no value of $x$ that makes it undefined (no denominator, no root that could be negative). So

Range: whether $x$ is positive, negative or zero, $|x|$ is always the non-negative value of $x$ — it can never come out negative, and it does attain every non-negative value (for instance, $|5|=5$ and $|-5|=5$ both reach $5$; $|0|=0$ reaches $0$). So

$\text{Range} = [0, \infty)$ …

A frequent mistake is writing the range as all of $\mathbb{R}$, forgetting that the whole point of the modulus is that it never produces a negative value — …