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Piecewise Functions

Different rules on different intervals — one graph, many pieces.

Interactive graph

Interactive Lab

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Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualEach piece is its own function — but only on a specific x-interval. Open dots mean 'not included'; closed dots mean 'included'.

Algebra $f(x) = \begin{cases} \text{rule}_1 & \text{condition}_1 \\ \text{rule}_2 & \text{condition}_2 \end{cases}$

f(x) = \begin{cases} x+2 & x<0 \\ x^2 & 0\le x\le 2 \\ 4 & x>2 \end{cases}

VisualAt every boundary, check: do the pieces meet? A jump = discontinuity.

AlgebraFor continuity at $x=c$ : $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = f(c)$ .

VisualFind which interval the input falls into, then use that piece's rule.

AlgebraTo find $f(1)$ in the example: $1 \in [0,2]$ , so $f(1) = 1^2 = 1$ .

Piecewise notation

f(x)=\begin{cases} \dots & \dots \\ \dots & \dots \end{cases}

Rule per interval

Continuity check

\lim_{x\to c^-} f = \lim_{x\to c^+} f = f(c)

At each boundary

Mark this lesson complete to update your mastery.

easypiecewise

Evaluate

f(2)

for

f(x)=\begin{cases} x+1 & x<2 \\ x^2 & x \ge 2 \end{cases}

easypiecewise

Evaluate

f(-3)

for

f(x)=\begin{cases} 2x & x<0 \\ 5 & x \ge 0 \end{cases}

mediumpiecewise

For

f(x)=\begin{cases} x+2 & x<1 \\ 3 & x \ge 1 \end{cases}

, is

f

continuous at

x=1

mediumpiecewise

For

f(x)=\begin{cases} x^2 & x<0 \\ -x & x \ge 0 \end{cases}

, find

f(0)

and the left-hand limit as

x \to 0^-

hardpiecewise

Write a piecewise function for the absolute value

f(x)=|2x-6|