# Piecewise Functions

## A Function Can be in Pieces

We can create functions that behave differently based on the input (x) value.

A function made up of 3 pieces

### Example:

- when x is less than 2, it gives
**x**,^{2} - 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 we write it:

The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this:

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: |

What is h(−1)?

x is ≤ 1, so we use h(x) = 2, so **h(−1) = 2**

What is h(1)?

x is ≤ 1, so we use h(x) = 2, so **h(1) = 2**

What is h(4)?

x is > 1, so we use h(x) = x, so **h(4) = 4**

Piecewise functions let us make functions that do anything we want!

### Example: A Doctor's fee is based on the length of time.

- Up to 6 minutes costs $50
- Over 6 and up to 15 minutes costs $80
- Over 15 minutes costs $80 plus $5 per minute above 15 minutes

Which we can write like this:

You visit for 12 minutes, what is the fee? $80

You visit for 20 minutes, what is the fee? $80+$5(20-15) = $105

## 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