Q.Find the domain and range of the modulus function $f(x) = |x|$.

Step 1: By its piecewise definition, $|x|=x$ for $x\ge0$ and $|x|=-x$ for $x<0$; either formula is defined for every real $x$, so domain $=\mathbb{R}$.

Step 2: In both cases $|x|\ge0$, and every non-negative real number $k$ is attained (at $x=k$, since $|k|=k$ for $k\ge0$).

Step 3: So the range is exactly the set of non-negative reals.

Confirm both pieces of the definition are defined for all reals, then argue every non-negative value (and no negative value) is attained.

• Reporting the range as $\mathbb{R}$, forgetting that $|x|$ can never be negative.
• Restricting the domain, thinking the modulus function is undefined for negative $x$ (it is defined — its value is simply $-x$ there).