The Range

The range is the set of outputs a function actually produces. For $f\left(x\right) = {x}^{2}+3$, the square ${x}^{2}$ of a real number is never negative, so the smallest output comes at x = 0 and is 3. Every number from 3 upward is reached, so the range is $y \ge 3$.

The same reasoning works for an absolute value. $g\left(x\right) = \left|x\right|-2$ has $\left|x\right| \ge 0$, so the outputs start at -2 and go up: the range is $y \ge -2$.

A non-horizontal line is different: it goes on forever up and down, so its range is all real numbers. A horizontal line y = 4 has a range of just one number, 4.