A piecewise function is one function written with more than one rule, each rule in charge of part of the number line. The rule is chosen by the input: $f(x) = \begin{cases} x + 1 & x < 2 \\ 3x - 5 & x \ge 2 \end{cases}$, read aloud: f of x equals x plus one when x is less than two, and three x minus five when x is at least two. Here $f$ is the function, $x$ is the input, and the conditions on the right decide which rule does the work. To evaluate $f(5)$, find the condition that 5 satisfies ($5 \ge 2$), then use that rule and no other: $3(5) - 5 = 10$. The other rule is not consulted, not averaged in, not added.

The one place to be careful is the boundary, the x where the rules change. Exactly one rule owns it, and the inequality signs say which: $\le$ or $\ge$ includes the boundary, $<$ or $>$ stops just short of it. On a graph that is the difference between a closed dot (the point is on the graph) and an open dot (the piece ends there without including it). In the example, $f(2) = 3(2) - 5 = 1$: the line $x + 1$ arrives at height 3 with an open dot, and the line $3x - 5$ starts at height 1 with a closed dot. Two heights at one x is what a jump looks like, and the dots say which height the function actually takes.