Answer: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:
Piecewise Function
(a solid dot means "including",
an open dot means "not including")
And this is how we write it:
Piecewise Function
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 member of Reals | 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:
continuous jump fn h(x) = 2 if x<=1, x if x>1
which looks like: continuous jump graph h(x)
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:
function piecewise f(t)=50 if t<=6, 80 if t>6 and t<=15, 80 + 5(t-15) if t>15
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
Absolute Value function
f(x) = |x|
Absolute Value function
The Floor Function
The Floor Function is a very special piecewise function. It has an infinite number of pieces:
Floor function
The Floor Function
