Piecewise Functions

Key Terms

What is a Piecewise Function

Piecewise Notation

Evaluating Piecewise Functions

Graphing Piecewise Functions

Domain of Piecewise Functions

Range of Piecewise Functions

Continuity of Piecewise Functions

Absolute Value as Piecewise

Step Functions

Real-World Applications

Writing Piecewise Functions

Common Mistakes

Workflow Summary

Piecewise Functions FAQ

Different Rules for Different Regions

Not every function follows a single formula across its entire domain. Tax rates change at income thresholds. Shipping costs jump at weight limits. A ball's height follows one equation while rising and another while falling. These situations demand functions that switch rules depending on where the input lands.

A piecewise function divides its domain into intervals, assigning a different formula to each piece. The function is defined everywhere it needs to be, but no single expression covers all cases. Instead, a collection of expressions — each with its own territory — combines to form one coherent function.

Key Terms

Core

Piecewise Function— multiple formulas, each governing a different part of the domain

Domain— the union of all intervals covered by the pieces

Range— the union of each piece's outputs on its interval

Related Types

Absolute Value Function— the most familiar piecewise function, with two linear pieces meeting at a vertex

Constant Function— each piece of a step function is constant

See All Function Definitions →

What is a Piecewise Function

A piecewise function is defined by multiple formulas, each applying to a different part of the domain. Rather than one rule governing all inputs, separate rules govern separate intervals.

The function remains a function — each input still produces exactly one output. But which formula produces that output depends on where the input falls.

A simple example:

f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}

For negative inputs, use

x + 1

. For non-negative inputs, use

x^2

. At

x = -3

, the output is

-3 + 1 = -2

. At

x = 2

, the output is

2^2 = 4

. At

x = 0

, the boundary, the condition

x \geq 0

applies, giving

0^2 = 0

.

Each piece can be any type of function — linear, quadratic, constant, radical, or otherwise. The pieces join together at boundary points, sometimes connecting smoothly, sometimes jumping abruptly.

One function, two formulas

The section's example: x + 1 for x < 0 (blue) and x² for x ≥ 0 (navy). Each input falls in exactly one interval, so exactly one formula runs: f(−3) = −2 from the first, f(2) = 4 from the second. At the boundary x = 0 the condition x ≥ 0 applies, so f(0) = 0; the first piece’s end at (0, 1) is open and never used.

Which formula applies is decided by the input, never by choice.

Piecewise Notation

Notation

Piecewise Notation

One function, several formulas — held together by a single brace, split apart by a boundary handshake, and drawn with a two-dot code. The grammar rules (each input matches exactly one row) are what make the stack a function at all.

f(x)

and evaluation come from function notation; the inequality signs and chains in the conditions from inequality notation; the intervals they encode from domain notation.

\begin{cases} 2x & x < 0 \\ 5 & x \geq 0 \end{cases}

f defined by cases

The single left brace stacks formula–condition rows and means "use the row whose condition holds". No right brace ever closes it — the layout itself, produced by the cases environment in LaTeX, is the notation.

CasesRows read "formula, if condition" — two rows or ten, the grammar is identical; the conditions must be mutually exclusive and cover the whole domain, or the stack stops defining a function.

Do not confuseSet braces. The same glyph collects members in

\{2, 5, 7\}

— there a brace pair encloses a finite set; here one unpaired brace selects a row. Twins in shape, strangers in meaning.

x < 1

x \geq 1

x below one; x at least one

The boundary handshake: adjacent conditions split each cut point so it belongs to exactly one row — strict on one side, or-equal on the other. Which row owns the boundary is a decision the inequality symbols record.

CasesCompound conditions chain —

1 \leq x < 4

, the chained inequality doing interval work inline; at evaluation time the handshake picks the piece, the drill of Evaluating Piecewise Functions below.

Do not confuseSymmetric inclusion. Writing

x \leq 1

and

x \geq 1

on adjacent rows double-books the boundary — legal only when both formulas agree there, a silent contradiction when they do not.

\bullet

\circ

solid dot in, open dot out

The graphical half of the handshake: a solid dot marks an included endpoint, an open circle an excluded one — the number-line code of domain notation, promoted to curve endpoints in Graphing Piecewise Functions below.

CasesAt each jump of a discontinuous graph, one solid and one open dot share a vertical line — the picture of exactly-one-row membership; two solid dots stacked vertically draw something that fails to be a function.

Do not confuseThe composition ring. The open dot on a graph is a hole, not the

\circ

of composition — position separates them: one sits on a curve, the other between function names.

All function symbols →↑ Functions

Evaluating Piecewise Functions

To evaluate a piecewise function at a specific input:

Step 1: Determine which condition the input satisfies.

Step 2: Use the corresponding formula to compute the output.

Only one piece applies to each input. The other formulas are irrelevant for that evaluation.

Let

f(x) = \begin{cases} 3x + 2 & \text{if } x < 1 \\ x^2 - 1 & \text{if } x \geq 1 \end{cases}

Find

f(-2)

: Since

-2 < 1

, use

3x + 2

f(-2) = 3(-2) + 2 = -4

.

Find

f(1)

: Since

1 \geq 1

, use

x^2 - 1

f(1) = 1 - 1 = 0

.

Find

f(4)

: Since

4 \geq 1

, use

x^2 - 1

f(4) = 16 - 1 = 15

.

Boundary points require attention. The condition that includes the boundary (with

\leq

\geq

) determines which formula to use. At

x = 1

above, the condition

x \geq 1

applies, not

x < 1

.

Errors often occur at boundaries when the wrong piece is selected. Always check the inequality symbols carefully.

Graphing Piecewise Functions

To graph a piecewise function, graph each piece on its designated interval, then combine them.

Step 1: Identify each piece and its interval.

Step 2: Graph each piece as if it were defined everywhere, but only draw the portion within its interval.

Step 3: Mark endpoints appropriately — solid dot for included endpoints, open dot for excluded endpoints.

For

f(x) = \begin{cases} x + 3 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases}

:

The first piece is a line with slope

1

, drawn for

x < 1

. At

x = 1

, this piece gives

1 + 3 = 4

, but

x = 1

is excluded, so place an open dot at

(1, 4)

.

The second piece is a line with slope

2

, drawn for

x \geq 1

. At

x = 1

, this piece gives

2(1) - 1 = 1

, and

x = 1

is included, so place a solid dot at

(1, 1)

.

The graph shows two line segments that do not connect — there is a jump at

x = 1

from height

1

to (approaching) height

4

.

The visual reveals discontinuities. Pieces that connect smoothly indicate continuity at that boundary. Pieces that don't connect indicate a jump.

Graph each piece on its own interval

Each line is drawn as if it ran everywhere (faint dashes), then only its own interval is kept. At x = 1 the first piece would give 4, but x = 1 is excluded, so (1, 4) gets an open dot; the second piece gives 1 and includes x = 1, so (1, 1) gets a solid dot. The gap between the two dots is the jump.

Open and closed dots are part of the graph, not decoration: they say which piece owns each boundary.

Domain of Piecewise Functions

The domain of a piecewise function is the union of all intervals covered by its pieces.

If the pieces cover all real numbers without gaps, the domain is

(-\infty, \infty)

.

If the pieces leave gaps, the domain excludes those intervals.

For

f(x) = \begin{cases} x^2 & \text{if } x \leq -1 \\ \sqrt{x} & \text{if } x \geq 0 \end{cases}

:

The first piece covers

(-\infty, -1]

. The second covers

[0, \infty)

. The interval

(-1, 0)

is not covered by either piece.

Domain:

(-\infty, -1] \cup [0, \infty)

.

Additionally, each piece must be valid on its interval. A piece involving

\sqrt{x - 3}

[0, 5]

would require

x \geq 3

for the formula to work, potentially reducing the effective domain of that piece to

[3, 5]

.

The domain is determined by both the stated conditions and the natural restrictions of each formula. Both must be satisfied.

Two intervals, one domain with a gap

x² is used only for x ≤ −1 and √x only for x ≥ 0. Laying the two intervals on one line and keeping everything either piece covers gives (−∞, −1] ∪ [0, ∞); the inputs between −1 and 0 belong to no piece, so they are not in the domain.

Remember to intersect each interval with what its own formula allows.

Range of Piecewise Functions

The range of a piecewise function is the union of the ranges of all pieces, restricted to their respective intervals.

Each piece contributes outputs from its portion of the domain. The combined range collects all these outputs.

For

f(x) = \begin{cases} 2 & \text{if } x < 0 \\ x + 1 & \text{if } x \geq 0 \end{cases}

:

The first piece is constant at

2

for all

x < 0

. It contributes

\{2\}

to the range.

The second piece is linear starting at

x = 0

. At

x = 0

, output is

1

. As

x \to \infty

, output

\to \infty

. This piece contributes

[1, \infty)

to the range.

Combined range:

\{2\} \cup [1, \infty) = [1, \infty)

(since

2

is already in

[1, \infty)

).

Finding the range may require analyzing each piece separately — finding minima, maxima, and behavior on each interval — then combining results.

Discontinuities can create gaps in the range if no piece produces certain values, or they can be invisible in the range if different pieces cover those values.

Collecting the outputs of each piece

The constant piece outputs only 2; the line x + 1, starting at (0, 1) with a solid dot, outputs every value from 1 upward. The amber bar on the y-axis is the union of the two: 2 already lies in [1, ∞), so the range is [1, ∞).

A piece's outputs can hide inside another piece's range, as the constant 2 does here.

Continuity of Piecewise Functions

A piecewise function is continuous at a boundary point if the two pieces meeting there produce the same value. The graph connects without a jump.

For continuity at

x = c

where two pieces meet:

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

The left piece's limit, the right piece's limit, and the actual value at

c

must all agree.

For

f(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 4x - 4 & \text{if } x > 2 \end{cases}

:

At

x = 2

: Left piece gives

2^2 = 4

. Right piece as

x \to 2^+

gives

4(2) - 4 = 4

. Both equal

4

.

The function is continuous at

x = 2

. The pieces connect.

For

g(x) = \begin{cases} x + 1 & \text{if } x < 3 \\ 2x - 3 & \text{if } x \geq 3 \end{cases}

:

At

x = 3

: Left piece as

x \to 3^-

gives

3 + 1 = 4

. Right piece gives

2(3) - 3 = 3

.

The limits differ (

4 \neq 3

), so the function is discontinuous at

x = 3

. There is a jump.

The table below collects both cases — the continuous example and the discontinuous example — with the three values that must agree for continuity and the verdict at each boundary.

Function and boundary	Left limit (x → c⁻)	Right limit (x → c⁺)	f(c)	Continuous at c?
x² (x ≤ 2), 4x − 4 (x > 2); at c = 2	4	4	4	✓
x + 1 (x < 3), 2x − 3 (x ≥ 3); at c = 3	4	3	3	✗

x + 1 for x < 0, x − 1 for x ≥ 0: a jump at 0

The two pieces meet the boundary x = 0 at different heights, 1 from the left and −1 from the right, so the graph jumps; the open dot marks the value the left piece approaches but does not take, the closed dot the value the function actually has. Continuity at the boundary would need the two dots to coincide. Move the pieces until they meet on the piecewise function explorer.

Checking continuity at a boundary is therefore a matter of comparing two limits with one value.

Absolute Value as Piecewise

The absolute value function is the most familiar piecewise function:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

For non-negative inputs, the output equals the input. For negative inputs, the output equals the negation of the input (making it positive).

The graph is V-shaped with vertex at the origin. The right arm has slope

1

; the left arm has slope

-1

. The pieces meet at

(0, 0)

and connect continuously — both pieces give

0

x = 0

.

Transformed absolute value functions shift the V:

f(x) = |x - 3| + 2 = \begin{cases} (x - 3) + 2 & \text{if } x \geq 3 \\ -(x - 3) + 2 & \text{if } x < 3 \end{cases} = \begin{cases} x - 1 & \text{if } x \geq 3 \\ -x + 5 & \text{if } x < 3 \end{cases}

The vertex moves to

(3, 2)

.

Equations involving absolute values often require piecewise analysis. Solving

|x - 2| = 5

means considering both pieces:

x - 2 = 5

(giving

x = 7

) and

-(x - 2) = 5

(giving

x = -3

|x| as two pieces: −x then x

The left ray is the line y = −x on x < 0 and the right ray is y = x on x ≥ 0; the two pieces meet at the origin at the same height, so the V is continuous even though its formula changes there. The closed dot at 0 belongs to the right piece. Rebuild the V from its two pieces on the piecewise function explorer.

Most functions with a corner are piecewise in this same quiet way.

Step Functions

Step functions are piecewise functions where each piece is constant. The graph consists of horizontal segments with jumps between them — like a staircase.

The floor function

\lfloor x \rfloor

returns the greatest integer less than or equal to

x

\lfloor 2.7 \rfloor = 2, \quad \lfloor -1.3 \rfloor = -2, \quad \lfloor 5 \rfloor = 5

The graph has horizontal segments at each integer height, jumping up at each integer input. The segment at height

n

covers

[n, n+1)

— closed on the left, open on the right.

The ceiling function

\lceil x \rceil

returns the least integer greater than or equal to

x

\lceil 2.7 \rceil = 3, \quad \lceil -1.3 \rceil = -1, \quad \lceil 5 \rceil = 5

The segments cover

(n-1, n]

— open on the left, closed on the right.

Both functions have jump discontinuities at every integer. The domain is all real numbers; the range is all integers

\mathbb{Z}

.

Step functions model situations with discrete jumps: postage rates by weight, tax brackets, rounding rules. The table below pairs the floor and ceiling functions side by side to highlight the small but important differences between them.

Function	Definition	Segment shape	Examples
Floor ⌊x⌋	greatest integer ≤ x	horizontal segment on [n, n+1) at height n — closed left, open right	⌊2.7⌋ = 2, ⌊−1.3⌋ = −2, ⌊5⌋ = 5
Ceiling ⌈x⌉	least integer ≥ x	horizontal segment on (n−1, n] at height n — open left, closed right	⌈2.7⌉ = 3, ⌈−1.3⌉ = −1, ⌈5⌉ = 5

The Heaviside step: −1 then 1

Each piece is a constant, so the graph is two horizontal segments with a jump of 2 at x = 0; the open and closed dots record which piece owns the boundary. Every step function is built this way, one flat piece per interval. Try the three-piece sign function on the piecewise function explorer.

Step functions model anything that changes only at thresholds.

Real-World Applications

Piecewise functions naturally model situations where rules change at thresholds.

Tax brackets: Income tax rates increase at certain income levels. Income below

\

10{,}000

might be taxed at

10\%

, income from

10{,}000

\

40{,}000

at

20\%

, and income above

40{,}000

30\%

T(x) = \begin{cases} 0.10x & \text{if } x \leq 10000 \\ 1000 + 0.20(x - 10000) & \text{if } 10000 < x \leq 40000 \\ 7000 + 0.30(x - 40000) & \text{if } x > 40000 \end{cases}

Shipping costs: Packages under

1

pound cost

\

. Packages from

to

pounds cost

5

plus

\

per additional pound. Packages over

pounds cost

13

plus

\

1.50$ per additional pound.

Utility rates: Electricity might cost one rate for the first

500

kWh and a higher rate beyond that threshold.

Parking fees: First hour free, next two hours at one rate, additional hours at another rate.

In each case, the rule changes at specific boundaries, and the piecewise function captures this structure mathematically. The table below collects four common application patterns with their threshold structure.

Situation	Threshold(s)	Piecewise structure
Tax brackets	income levels (e.g., $10,000, $40,000)	tax rate increases at each bracket boundary; pieces are continuous (rates align at thresholds)
Shipping costs	weight bands (under 1 lb, 1–5 lb, over 5 lb)	flat fee for first band, then per-unit surcharge added above
Utility rates	usage tier (e.g., 500 kWh)	one rate up to the tier, a higher rate beyond
Parking fees	time periods (first hour, next two, beyond)	different rate per time period; often a free or fixed first piece

Writing Piecewise Functions

Constructing a piecewise function from a description or graph requires identifying the pieces and their intervals.

From a graph:

Step 1: Identify distinct segments or curves.

Step 2: Determine the formula for each segment (line, parabola, constant, etc.).

Step 3: Identify the interval each segment covers, noting open or closed endpoints.

Step 4: Write the piecewise notation with each formula and its condition.

From a verbal description:

Step 1: Identify the thresholds where rules change.

Step 2: Write the formula for each region between thresholds.

Step 3: Specify conditions using inequalities.

Example: "A plumber charges

\

for the first hour and

30

for each additional hour."

Let

C(t)

be the cost for

t

hours of work (

t > 0

C(t) = \begin{cases} 50 & \text{if } 0 < t \leq 1 \\ 50 + 30(t - 1) & \text{if } t > 1 \end{cases}

Simplifying the second piece:

50 + 30t - 30 = 30t + 20

C(t) = \begin{cases} 50 & \text{if } 0 < t \leq 1 \\ 30t + 20 & \text{if } t > 1 \end{cases}

The plumber's charge as a graph

C(t) = 50 for 0 < t ≤ 1 and 30t + 20 for t > 1. The flat piece is the first-hour charge; after the threshold at one hour, the line climbs $30 per hour. Both pieces give $50 at t = 1, so the graph has no jump, and three hours cost C(3) = $110.

Drawing the function is a quick check that the written pieces match the description.

Common Mistakes

Several errors recur when working with piecewise functions.

Using the wrong piece at a boundary: At

x = 3

with conditions

x < 3

and

x \geq 3

, the second piece applies, not the first. The inequality symbols determine which piece owns the boundary.

Overlapping conditions: Writing

x \leq 2

and

x \geq 2

creates overlap at

x = 2

. Both pieces claim this input, making the function ambiguous. Use

x < 2

and

x \geq 2

, or

x \leq 2

and

x > 2

.

Gaps in conditions: Writing

x < 1

and

x > 3

leaves

[1, 3]

undefined. Unless this gap is intentional (limiting the domain), conditions should cover all intended inputs.

Assuming continuity: Not all piecewise functions are continuous. Checking whether pieces connect at boundaries is essential — do not assume they do.

Forgetting endpoint markers when graphing: Open and closed dots distinguish included from excluded endpoints. Without them, the graph is incomplete and potentially misleading.

Evaluating all pieces instead of one: Only the piece whose condition the input satisfies should be used. The other formulas do not apply to that input.

The table below collects each mistake with the specific problem it causes and the rule of thumb that prevents it.

Mistake	What goes wrong	How to avoid it
Using the wrong piece at a boundary	wrong output at boundary inputs	check inequality symbols — the piece with ≤ or ≥ owns the boundary
Overlapping conditions	function is ambiguous where conditions overlap	pair ≤ with >, or < with ≥, so each boundary lives in exactly one piece
Gaps in conditions	function undefined on some intervals	verify conditions cover the entire intended domain
Assuming continuity	misreading behavior at boundaries	always check that adjacent pieces produce the same value at the boundary
Forgetting endpoint markers	graph is incomplete or misleading	solid dot for included endpoint, open dot for excluded — for every piece transition
Evaluating all pieces at one input	wrong answer or confusion about which output applies	only the piece whose condition the input satisfies should be used

Workflow Summary

Most of the tasks on this page — evaluating a piecewise function at an input, graphing it, finding its domain or range, checking continuity at a boundary — share the same general principle: handle each piece on its own interval, then combine. The table below collects the five core tasks with the procedure for each and the pitfall to watch for, useful as a checklist when working through a new piecewise function from scratch.

Task	What to do	Watch out for
Evaluate at a specific input	identify which condition the input satisfies; apply that formula only	boundary points — the inequality symbols decide which piece owns the boundary
Graph the function	graph each piece on its interval and combine	use solid dots for included endpoints, open dots for excluded
Find the domain	take the union of the intervals stated by the conditions	also check each formula's natural restrictions (denominators, square roots, logs)
Find the range	find the outputs of each piece on its interval, then take the union	gaps in the range can appear if no piece covers some y-values
Check continuity at a boundary c	compute left limit, right limit, and f(c)	all three must agree — any mismatch is a discontinuity

Piecewise Functions FAQ

How do you evaluate a piecewise function at a point?

Find which condition the input satisfies, then use only that branch. Check the inequalities carefully at boundary values, since one branch usually includes the endpoint and the other excludes it. Substituting into the wrong piece, or into several pieces at once, is the usual mistake.Read more →

When do you use an open circle and when a filled one?

A filled circle marks an endpoint the branch includes, matching a condition written with less-than-or-equal. An open circle marks one it excludes, matching a strict inequality. At any boundary exactly one branch owns the point, so exactly one circle is filled, which is what keeps the rule single-valued.Read more →

How do you check whether a piecewise function is continuous?

Look only at the boundaries, since each branch is usually continuous on its own. At each boundary, evaluate the branch approaching from the left and the branch approaching from the right. Matching values mean the graph connects; different values mean a jump. A single value defined there is not enough on its own.Read more →

How is the absolute value function a piecewise function?

It is defined by two branches: the input itself when it is non-negative, and the negated input when it is negative. This is why the graph is a V, with the corner sitting where the two branches meet at zero. Rewriting absolute values this way is how equations containing them are usually solved.Read more →

What is the domain of a piecewise function?

The union of the intervals its branches cover, minus any gap none of them account for. If the conditions leave a value unassigned the function is simply undefined there. Overlapping conditions are worse: they assign two outputs to one input, which stops the rule being a function at all.Read more →