1. Piecewise Functions and Step Functions

2. What Are They?
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.
Piecewise Function –a function defined by two or more functions over a specified domain.

3. f(x) = What do they look like? x2 + 1 , x  0 x – 1 , x  0
You can EVALUATE piecewise functions.
You can GRAPH piecewise functions.

4. When do we use them in real life?
All the time.
Here is one example:
Admission fees. A local zoo charges admission to groups according to the following policy. Groups of fewer than 50 people are charged a rate of per person, while groups of 50 people or more are charged a reduced rate of per person.
This situation can be represented by a piecewise function. We will come back to this example at the end of the lesson.

5. f(x) = II. Evaluating Piecewise Functions: Let’s calculate f(2).
Evaluating piecewise functions is just like evaluating functions that you are already familiar with.
Let’s calculate f(2).
f(x) =
x , x  0
x – 1 , x  0
You are being asked to find y when
x = 2. Since 2 is  0, you will only substitute into the second part of the function.
f(2) = 2 – 1 = 1

6. f(x) = Let’s calculate f(-2). x2 + 1 , x  0 x – 1 , x  0
You are being asked to find y when
x = -2. Since -2 is  0, you will only substitute into the first part of the function.
f(-2) = (-2)2 + 1 = 5

7. f(x) = Your turn: 2x + 1, x  0 2x + 2, x  0 Evaluate the following:? -3
f(5) = 12 ?
f(1) = 4 ?
f(0) = ? 2

8. f(x) = One more: 3x - 2, x  -2 -x , -2  x  1 x2 – 7x, x  1
Evaluate the following:
f(-2) = ? 2
f(3) =
-12 ?
f(-4) =
-14 ? ?
f(1) = -6

9.  f(x) = III. Graphing Piecewise Functions: x2 + 1 , x  0
Determine the shapes of the graphs.
Parabola and Line
Determine the boundaries of each graph.    
Graph the parabola where x is less than zero.
Graph the line where x is greater than or equal to zero.
Notice the closed vs open circles. 
Domain:
Range:

10.   f(x) = Graphing Piecewise Functions: 3x + 2, x  -2
Determine the shapes of the graphs.
Line, Line, Parabola
Determine the boundaries of each graph.        

11. IV. Applications
Admission fees. A local zoo charges admission to groups according to the following policy. Groups of fewer than 50 people are charged a rate of per person, while groups of 50 people or more are charged a reduced rate of per person.
Find a mathematical model expressing the amount a group will be charged for admission as a function of its size.

13. Notes: Step Functions

14. I. What is it?
A step function looks like a steps on a staircase. They can be represented by a piecewise function, or the greatest integer function. Try graphing the following piecewise function.

16. Try another:

18. II. Special Step Functions
Two particular kinds of step functions are called ceiling functions ( f (x)= ]x[ and floor functions (f (x)=[x]).
Ceiling Functions:
In a ceiling function, all nonintegers are rounded up to the nearest
integer. This is also called the ‘least integer function’.
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.

19. Least Integer Function:

20. Least Integer Function:

21. Least Integer Function:

22. Least Integer Function:

23. Least Integer Function:
Don’t worry, there are not wall functions, front door functions, fireplace functions!
The least integer function is also called the ceiling function.
The notation for the ceiling function is:
The TI-89 command for the ceiling function is ceiling (x).

24. B. Floor Function/Greatest Integer Function
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].

25. Greatest Integer Function:

26. Greatest Integer Function:

27. Greatest Integer Function:

28. Greatest Integer Function:

29. III. Applications of Step Functions
PSYCHOLOGY One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.
Understand The total charge must be a multiple of $85, so the graph will be the graph of a step function.
Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on.

30. Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph.

31. Answer:
Check Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint.

32. Try this!
SALES The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.
A. B.
C. D.

33. Homework You are to complete #60 and #61 tonight.
#60: Graphing Piecewise Functions
Skip #2
#61 Step Functions WS
Skip # 5