Piecewise Functions

Example: Imagine a function

• 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 we write it like this:

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:

• using Interval Notation:   Dom(f) = (-∞, 6]
• using Set Builder Notation:   Dom(f) = {x ∈ | x ≤ 6}

And here are some example values:

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

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