How to Graph Piecewise Functions

Last reviewed September 18, 2026

A piecewise function uses different rules on different parts of its domain. The mathematics is no harder than the pieces it is built from; what makes them fiddly is the boundaries, where a single filled or hollow circle changes the answer.

What the notation says

A piecewise definition is a list of rules, each with a condition:

f(x) = x + 3 when x < 1
f(x) = x² when x ≥ 1

Read it as an instruction. Given an x, check the conditions from the top, and use the first rule whose condition is satisfied. At x = 0 the first applies, giving 3. At x = 2 the second applies, giving 4.

The conditions must cover the whole domain and must not overlap. If two rules both claimed x = 1 and disagreed about the answer, it would not be a function at all.

Drawing one, step by step

Graph the function above.

1. Take one piece at a time. Ignore the conditions for a moment and picture y = x + 3 as a whole line, and y = x² as a whole parabola.
2. Keep only the part the condition allows. Draw the line only to the left of x = 1, and the parabola only from x = 1 rightwards.
3. Work out the boundary values. The line approaches 1 + 3 = 4; the parabola starts at 1² = 1.
4. Mark the endpoints. The condition on the line is x < 1, strict, so (1, 4) gets an open circle. The condition on the parabola is x ≥ 1, so (1, 1) gets a filled one.
5. Leave the gap. The function jumps from near 4 to 1 at x = 1, and nothing is drawn in between.

Open and closed endpoints

This is where most of the marks are, and the rule is short:

At every boundary exactly one of the two pieces owns the point, so you should end up with exactly one filled circle and, if the pieces disagree, one open circle directly above or below it.

Continuous, jump and removable

What happens at the boundary has a name, and questions often ask for it.

Continuous

Both pieces arrive at the same value, so the graph joins smoothly and you could draw it without lifting the pen. For f(x) = 2x when x < 3 and f(x) = x + 3 when x ≥ 3, both give 6 at the boundary. Continuous.

Jump discontinuity

The two pieces arrive at different values, as in the first example. The function is defined at the boundary — the filled circle says which value it takes — but it leaps.

Removable discontinuity

Both pieces approach the same value, but the function is defined as something else at that single point, or not defined at all. The graph looks continuous apart from one misplaced dot. It is called removable because redefining that one point would fix it.

Typing one into the calculator

There are two ways, and both are worth knowing.

Braces with conditions

Write the whole thing in one expression:

{x < 1: x + 3, x >= 1: x^2}

Each branch is a condition, a colon, then the value. A final branch with no condition acts as the catch-all: {x > 0: ln(x), -ln(-x)}.

One expression per piece

Often clearer while you are learning. Type each piece on its own line with a restriction attached:

(x + 3){x < 1} on one line, x^2{x >= 1} on the next.

A restriction in braces evaluates to 1 where the condition holds and to undefined elsewhere, so multiplying by it keeps the curve only where you want it. As a bonus, each piece gets its own colour, which makes the structure obvious.

Where they turn up in real problems

Piecewise functions are not an artificial exercise. They are how you model anything with a threshold.

• Tax brackets. A different rate applies to each band of income.
• Shipping costs. One price up to a weight, another above it.
• Phone plans. A flat fee until an allowance runs out, then a per-unit charge.
• Overtime pay. Standard rate to forty hours, higher rate beyond.

All four produce continuous piecewise functions, because the two rules agree at the threshold. A jump discontinuity in a pricing model usually signals a mistake in the pricing.

Questions

Build a piecewise function