How do I know which piece of a piecewise function to use?

Check the value of your input against the inequality condition in each piece. Find the piece whose condition is true, and apply that formula. For example, if x = -3 and one piece has x > 0, that condition is false so do not use that piece; if another has x \leq 0, that condition is true so use it.

Why do piecewise functions have different formulas?

Real situations often have different rules in different contexts: a phone plan charges one rate for the first 1000 minutes and another rate beyond that, or temperature might rise at one rate during the day and fall at a different rate at night. Piecewise functions model these natural changes in behavior.

Can a piecewise function be continuous at the boundary?

Yes, it is possible. If the output of one piece at the boundary equals the output of the next piece, then the function is continuous at that point and there is no jump. However, piecewise functions often have jump discontinuities by design, and this is perfectly correct—do not assume continuity unless the problem asks for it.

How do I graph a piecewise function?

Graph each piece separately over its own interval, then combine them. For the first interval, plot the first formula but only for x values in its domain; for the second interval, plot the second formula but only for its x values. Use open circles to show excluded endpoints (from ) and closed circles to show included endpoints (from \leq or \geq).

Piecewise Functions

A piecewise function uses different formulas for different input ranges, letting you define relationships that change behavior at specific boundaries.

f(x) = \begin{cases} f_1(x) & x \in D_1 \\ f_2(x) & x \in D_2 \end{cases}

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What each symbol means

What Piecewise Functions takes

Symbol	Meaning
$f$	The piecewise function itself, which consists of multiple sub-functions applied to different input regions; treating it as a single unchanging rule rather than recognizing its multi-part nature will cause you to apply the wrong formula for some inputs.
$x$	The input variable (independent variable) whose value determines which piece of the function to use; each value of $x$ belongs to exactly one of the domain regions $D_1, D_2$ , etc.

When to use it

Use a piecewise function when the rule or formula for a relationship changes depending on the value of the input.

Level

Usually taught in: Algebra II

Worked examples

1. Evaluate a piecewise function at multiple points

Problem

Evaluate

f(2)

and

f(-1)

where

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

$f(2) = 2(2) + 1$
Since $2 \geq 0$ , we use the first piece of the function.
$f(2) = 4 + 1$
Multiply: $2 \cdot 2 = 4$ .
$f(2) = 5$
Add: $4 + 1 = 5$ .
$f(-1) = -(-1)$
Since $-1 < 0$ , we use the second piece of the function.
$f(-1) = 1$
Negate: $-(-1) = 1$ .

Answer: $f(2) = 5 \text{ and } f(-1) = 1$

Piecewise functions require you to first identify which interval contains your input, then apply the corresponding formula; the same function $f$ gives different outputs for different inputs because they use different rules.

2. Evaluate with negative input and check boundary conditions

Problem

Find

f(0)

and

f(-3)

where

f(x) = \begin{cases} \frac{1}{2}x + 2 & x > 0 \\ -x^2 & x \leq 0 \end{cases}

$f(0) = -(0)^2$
Since $0 \not> 0$ but $0 \leq 0$ , we use the second piece $-x^2$ .
$f(0) = -(0)$
Square: $0^2 = 0$ .
$f(0) = 0$
Negate: $-(0) = 0$ .
$f(-3) = -(-3)^2$
Since $-3 \leq 0$ , we use the second piece $-x^2$ .
$f(-3) = -(9)$
Square: $(-3)^2 = 9$ .
$f(-3) = -9$
Negate: $-(9) = -9$ .

Answer: $f(0) = 0 \text{ and } f(-3) = -9$

Piecewise functions require careful attention to boundary conditions and inequality symbols; here, the boundary value $x = 0$ belongs to the second piece because of the condition $x \leq 0$ (which includes equality), not the first piece with $x > 0$ .

3. Model temperature changes in a laboratory

Problem

A laboratory measures temperature during an experiment. From

t = 0

t = 4

hours, temperature rises from 20°C at a rate of 3°C per hour. From

t = 4

t = 8

hours, temperature falls at a rate of 2°C per hour. Write the piecewise temperature function

T(t)

and find the temperature at

t = 2

hours and

t = 6

hours.

$T(t) = \begin{cases} 20 + 3t & 0 \leq t \leq 4 \\ 40 - 2t & 4 < t \leq 8 \end{cases}$
For the first interval, temperature starts at 20°C and increases 3°C per hour. At $t = 4$ , we have $20 + 3(4) = 32$ °C. For the second interval starting from 32°C and decreasing 2°C per hour, the formula is $32 - 2(t - 4) = 40 - 2t$ .
$T(2) = 20 + 3(2)$
Since $2$ is in the interval $0 \leq t \leq 4$ , we use the first formula.
$T(2) = 20 + 6$
Multiply: $3 \cdot 2 = 6$ .
$T(2) = 26$
Add: $20 + 6 = 26$ .
$T(6) = 40 - 2(6)$
Since $6$ is in the interval $4 < t \leq 8$ , we use the second formula.
$T(6) = 40 - 12$
Multiply: $2 \cdot 6 = 12$ .
$T(6) = 28$
Subtract: $40 - 12 = 28$ .

Answer: $T(2) = 26 \text{°C and } T(6) = 28 \text{°C}$

Real-world situations like temperature change over time often involve different rates of change over different time periods; piecewise functions are the natural way to model these changes by defining separate formulas for each time interval.

Common mistakes

Where Piecewise Functions usually goes wrong

The mistake	Why it is wrong	The fix
$f(-2) = 2(-2) + 1 = -4 + 1 = -3$ when evaluating $f(x) = \begin{cases} 2x + 1 & x \geq 0 \\ x & x < 0 \end{cases}$	This student applied the first piece (valid for $x \geq 0$ ) to the input $x = -2$ , which actually belongs to the second piece where $x < 0$ .	Before applying any formula, check which interval contains your input: does $-2 \geq 0$ ? No. Does $-2 < 0$ ? Yes. So use the second piece: $f(-2) = -2$ .
Using the same formula for both $x = 5$ and $x = 5.1$ when the first piece is defined as $x < 5$ .	Confusing inequality symbols; the symbol $<$ does not include the boundary value, so $x = 5$ does not satisfy $x < 5$ .	Check the exact boundary inequality: $<$ means the boundary is excluded (so 5 does not belong to $x < 5$ ), while $\leq$ means it is included (so 5 does belong to $x \leq 5$ ). Always match your input to the exact inequality.
Assuming the two pieces must connect at the boundary and treating the function as 'wrong' if they do not.	Piecewise functions are designed precisely to model situations where behavior changes abruptly; a jump discontinuity (where pieces do not connect) is intentional and expected in many real-world applications like tax brackets or price changes.	Do not assume continuity unless the problem explicitly requires it; verify by checking whether the output of one piece at the boundary equals the output of the adjacent piece, but understand that a jump is normal and correct in piecewise functions.

Tips and when to use something else

Always identify which interval your input belongs to before applying any formula; this single step prevents most errors with piecewise functions.
Pay close attention to inequality symbols: $<$ and $>$ exclude the boundary, while $\leq$ and $\geq$ include it. One wrong symbol means you apply the wrong piece.
If a function can be written as a single formula without branches, write it that way instead; piecewise notation is for situations where the rule genuinely changes.
Jump discontinuities (where pieces do not connect at boundaries) are normal and intentional in piecewise functions; they model real situations like price brackets or tax rates that change abruptly.

Related formulas

Function NotationFunction notation y = f(x) is a way to write and evaluate functions, showing how input values transform into output values using a concise symbolic format.Domain and RangeDomain lists all valid inputs for a function; range lists all possible outputs—use this to understand what a function can do.Absolute Value EquationsSolve equations with absolute value by splitting into two cases where the expression inside equals the positive value or its negative.Function TransformationsFunction transformations describe how shifting, stretching, and flipping graphs of functions create new related functions from a parent function.

Frequently asked questions

How do I know which piece of a piecewise function to use?: Check the value of your input against the inequality condition in each piece. Find the piece whose condition is true, and apply that formula. For example, if $x = -3$ and one piece has $x > 0$ , that condition is false so do not use that piece; if another has $x \leq 0$ , that condition is true so use it.
Why do piecewise functions have different formulas?: Real situations often have different rules in different contexts: a phone plan charges one rate for the first 1000 minutes and another rate beyond that, or temperature might rise at one rate during the day and fall at a different rate at night. Piecewise functions model these natural changes in behavior.
Can a piecewise function be continuous at the boundary?: Yes, it is possible. If the output of one piece at the boundary equals the output of the next piece, then the function is continuous at that point and there is no jump. However, piecewise functions often have jump discontinuities by design, and this is perfectly correct—do not assume continuity unless the problem asks for it.
How do I graph a piecewise function?: Graph each piece separately over its own interval, then combine them. For the first interval, plot the first formula but only for $x$ values in its domain; for the second interval, plot the second formula but only for its $x$ values. Use open circles to show excluded endpoints (from $<$ or $>$ ) and closed circles to show included endpoints (from $\leq$ or $\geq$ ).

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Reviewed 2026-09-18