1. Piecewise Functions

2. Up to now, we’ve been looking at functions represented by a single equation.
In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.
These are called piecewise functions.

3. One equation gives the value of f(x) when x ≤ 1
And the other when x>1

4. Evaluate f(x) when x=0, x=2, x=4
First you have to figure out which equation to use
You NEVER use both
X=0
This one fits into the top equation
X=2
X=4
This one fits here
This one fits here
So:
2(2) + 1 = 5
f(2) = 5
0+2=2
f(0)=2
2(4) + 1 = 9
f(4) = 9

5. Graph: For all x’s < 1, use the top graph (to the left of 1)
For all x’s ≥ 1, use the bottom graph (to the
right of 1)

6. x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.

7. Graph:
Point of Discontinuity

8. You have a summer job that pays time and a half for overtime
You have a summer job that pays time and a half for overtime. That is, if you work more than 40 hours per week, your hourly wage for the extra hours is 1.5 times your normal hourly wage of $7.00.
Write and a graph a piecewise function that gives you your weekly pay P in terms of the number of hours h you work.
How much will you get paid if you work 45 hours?

9. GARAGE RATES $3 per half hour $8 maximum for 12 hours
Write a piecewise function for the parking garages shown on the sign.
What is the domain and range of the function?

10. Step Functions

12. Graph :

14. Special Step Functions
Two particular kinds of step functions are called ceiling functions ( f (x)= and floor functions ( f (x)= ).
In a ceiling function, all nonintegers are rounded up to the nearest
integer.
An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes.

15. Special Step Functions
In a floor function, all nonintegers are rounded down to the nearest integer.
The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday.
The floor function is the same thing as the greatest integer function which can be written as f (x)=[x].