Piecewise Functions
A Function Can be in Pieces
You can create functions that behave differently depending on the input (x) value.
A function made up of 3 pieces
Example: A function with three pieces:
- when x is less than 2, it gives x2,
- when x is exactly 2 it gives 6
- when x is more than 2 and less than or equal to 6 it gives the line 10-x
It looks like this:
(a solid dot means "including",
an open dot means "not including")
And this is how you write it:
The Domain is all Real Numbers up to and including 6:
Dom(f) = (-∞, 6] (using Interval Notation)
Dom(f) = {x 
| x ≤ 6} (using Set Builder Notation)
And here are some example values:
| X | Y |
|---|---|
| -4 | 16 |
| -2 | 4 |
| 0 | 0 |
| 1 | 1 |
| 2 | 6 |
| 3 | 7 |
Example: Here is another piecewise function:
 |
which looks like: |  |
The Absolute Value Function
The Absolute Value Function is a famous Piecewise Function.
It has two pieces:
- below zero: -x
- from 0 onwards: x
f(x) = |x|
The Floor Function
The Floor Function is a very special piecewise function. It has an infinite number of pieces:
The Floor Function