1. Step Functions
Section 3.5

2. Objectives:
Define and graph the greatest integer and the rounding-up functions. Define step functions, and use them to model real-world applications.

3. Step Functions
Look like steps. It remains constant within a series of intervals but varies between intervals.

4. Step Functions
The greatest integer function is denoted by and is defined by:

5. 3 4 -6 The greatest integer function is an example of a step function.
Some examples of the values:
[3.5] =
[4.7] =
[-5.7] = 3 4 -6

6. Graph of the Greatest Integer Function:2 4 6 –2 –4 –6 x y
y=[x]

7. y=[x] The greatest integer less than or equal to x2 4 6 –2 –4 –6 x y
y=[x]
The greatest integer less than or equal to x
The domain is the set of real numbers.
The range is the set of integers.

8. We can write this in function notation as
Example 1:
Undergraduate Classification at Study-Hard University (SHU) is a function of Hours Earned.
We can write this in function notation as
C = f(H).

9. Classification of Students at SHU (From the Catalogue)
C = f(H)
No student may be classified as a sophomore until after earning at least 30 semester hours.
No student may be classified as a junior until after earning at least 60 hours.
No student may be classified as a senior until after earning at least 90 hours.
Evaluate f(20)
f(20) = Freshman
Evaluate f(30)
f(30) = Sophomore
Evaluate f(0)
f(0) = Freshman
Evaluate f(61)
f(61) = Junior

10. The domain of f is the set of non-negative integers
C = f(H)
For this function, What is the domain of f ?
The domain of f is the set of non-negative integers
What is the range of f?
Range of f is {Fr, Soph, Jr, Sr}

11. Diagram of C = f(H)

12. C = f(H)
Symbolic Representation

13. C = f(H) Graphical Representation
Note: that in this graph the domain is considered to be [0, ∞)

14. Classwork:
Step Functions Worksheet

15. Step Functions
The rounding up function, or least-integer function is denoted by and is defined by:
Example: Telephone calls are based on the minutes of the call. If you talk 2.5 minutes, they charge you for 3 minutes.

16. 3 -1 6 The rounding-up function is an example of a step function.
Some examples of the values: = 3 -1 6

17. Example 2:
Downtown Parking charges a $5 base fee for parking through 1 hour, and $1 for each additional hour or fraction thereof.
The maximum fee for 24 hours is $15.
Sketch a graph of the step function that describes this pricing scheme.

18. The maximum fee for 24 hours is $15.
Solution
Downtown Parking charges a $5 base fee for parking through 1 hour, and $1 for each additional hour or fraction thereof.
The maximum fee for 24 hours is $15.
Sample of ordered pairs (hours, price):
(.25,5), (.75,5), (1,5), (1.5,6), (1.75,6)

19. During the 1st hour: price = $5 During the 2nd hour: price = $6
During the 3rd hour: price = $7
During the 11th hour: price = $15
It remains at $15 for the rest of
the 24-hour period.
Plot the graph on the interval (0,24].

20. Equation for the last graph:

21. Homework:
Practice and Apply 3.5