Piecewise Functions

What is a Piecewise Function?

A piecewise function uses different rules for different parts of its domain.

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

This says:

• For negative inputs → square them
• For zero or positive inputs → use $2x + 1$

Evaluating Piecewise Functions

Step 1: Check which condition the input satisfies

Step 2: Use that rule

Example: Using the function above, find $f(-3)$

• Is $-3 < 0$? Yes
• Use the first rule: $f(-3) = (-3)^2 = 9$

Example: Find $f(2)$

• Is $2 < 0$? No
• Is $2 \geq 0$? Yes
• Use the second rule: $f(2) = 2(2) + 1 = 5$

Example: Find $f(0)$

• Is $0 < 0$? No
• Is $0 \geq 0$? Yes
• Use the second rule: $f(0) = 2(0) + 1 = 1$

Note: At boundaries, check carefully which condition includes the equals sign.

Do the Pieces Connect?

At the boundary point, do the two rules give the same output?

For our function at $x = 0$:

• Left piece: $0^2 = 0$
• Right piece: $2(0) + 1 = 1$

They don’t match. There’s a jump at $x = 0$.

Continuous vs. Discontinuous

Continuous: The pieces connect smoothly (no jumps or gaps)

Discontinuous: There’s a break at one or more points

Example of a continuous piecewise function:

$g(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$

At $x = 1$:

• Left piece: $1^2 = 1$
• Right piece: $2(1) - 1 = 1$

They match! This function is continuous.

The Absolute Value Function

The most common piecewise function:

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

• Positive numbers: stay the same
• Negative numbers: sign gets flipped

Examples:

• $|5| = 5$
• $|-3| = -(-3) = 3$
• $|0| = 0$

Key idea: Absolute value gives the distance from zero, which is always non-negative.

Writing Piecewise Functions

Sometimes you need to write your own.

Example: Write a function that outputs 0 for negative inputs and equals the input for non-negative inputs.

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

Common Patterns

Step function:

$f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases}$

This is the sign function (also written $\text{sgn}(x)$).

Tax brackets, shipping costs, and grading scales are all real-world piecewise functions.

Different rules apply depending on which “bracket” you fall into.