What is a function (quick reminder)?

A function is a rule that takes an input (usually $x$) and gives exactly one output (usually $y$).

Example:

f(x)=2x

If x=3, then f(3)=6

What does “piecewise” mean?

Piecewise means “made of pieces.”

A piecewise function uses different rules depending on the value of $x$.

Instead of one formula, you have multiple formulas, each used on a specific interval.

How piecewise functions are written

They usually look like this:

            {2x+1 if x<0

f(x)=

            {x^2 if x≥0​

This means:

• If $x$ is less than 0, use 2x+1

• If $x$ is 0 or greater, use x^2

How to evaluate a piecewise function

Step 1: Look at the value of x
Step 2: Decide which rule applies
Step 3: Plug $x$ into that rule

Example:
Find $)$

Since −2<0, use:

2x + 1 = 2(-2) + 1 = -3

So, f(-2) = -3

Why piecewise functions exist

They’re used when:

• Rules change (tax brackets, phone plans, shipping costs)

• Graphs behave differently in different regions

• Real-world situations aren’t “one-rule-fits-all”