Step Function

Key Concepts

• Understand Step Functions and its different types.
• Use a Step Function to represent a real-world situation.
• Use a Step Function to solve problems. 

Understand Step Functions 

1. What is the graph of the ceiling function?  

A step function is a piecewise-defined function that consists of constant pieces. The graph resembles a set of steps. 

The ceiling function is a kind of step function. It rounds numbers up to the nearest integer. It is notated as f(x) = ceiling(x) or f(x) = [ X ]. 

Make a table of values and graphs.  

The domain is all real numbers. The range is all integers.  

2. What is the graph of the floor function?  

The floor function is another kind of step function. It rounds numbers down to the nearest integer. It is notated as  

f(x) = floor(x) or f(x) = [ X ] 

Make a table of values and graph.   

The domain is all real numbers. The range is all integers. 

Use a Step Function to represent a real-world situation 

Some students are planning a field trip. If there are 40 students and adults or fewer going on the field trip, they rent vans that hold 15 people. If there are more than 40 students and adults, they rent buses that hold 65 people.  

1. What function can you use to represent this situation? 

2. How many buses are needed if 412 students and adults are going on a field trip? 

Evaluate the function for f(412). 

f(412) = ⌈412 / 65⌉

=⌈6.34⌉

= 7 

Seven buses are needed if 412 students and adults are going on a field trip.   

Use a Step Function to solve problems 

Jamal and his brother plan to rent a karaoke machine for a class event. The graph shows the rental costs.  

1. How much should they expect to spend if they rent the karaoke machine from 8 am until 7.30 pm? 

Step 1: Write a function to represent the rental costs.  

Step 2: Determine the duration of the rental. 

8 A.M. to 7.30 P.M. is 11 h, 30 min or 11 ½ h.  

Step 3: Evaluate the function for f(11.5).  

f(11.5) = 10 ceiling ⌈11.5⌉ + 20  

              = 10(12) + 20  

              = 140  

The cost of the rental will be $140.  

2. The class event ended early, so Jamal could return the machine by 7.05 PM. How much money would he save if he returned the machine at 7.05 PM?  

Jamal would save no money if he returned the machine at 7.05 PM. He will be charged for the

Questions  

Question 1 

Evaluate each function for the given value.  

1. f(x) = ceiling(x); x = 2.65  

2. f(x) = floor(x); x = 2.19  

Solution: 

1. f(x) = ceiling(2.65) = 3  

2. f(x) = floor(2.19) = 2  

Question 2 

The postage for a first-class letter weighing one ounce or less is $0.47. Each additional ounce is $0.21. The maximum weight of a first-class letter is 3.5 oz. Write a function to represent this situation.  

Solution:  

f(x) = 0.47ceiling(x), 0<x<=1  

When x>1 and x<=3.5,  

f(x) =  0.47*1 + 0.21 ceiling(x) – 0.21 = 0.26 + 0.21 ceiling(x)  

Function:  

f(x) = 0.47 ceiling(x), 0<x<=1 

f(x) = 0.26 + 0.21 ceiling(x), 1<x<=3.5  

Question 3 

In the example in section 1.3, you rent a karaoke machine at 1 PM and plan to return it by 4 PM. Will you save any money if you return the machine 15 min. early? Explain.  

Solution:  

Function is: 

From 1 PM to 4 PM, it’s 3 hours.  

f(x) = 10*3 + 20 = $50  

Now if I return the karaoke machine at 4.45 pm, the value of x is 2 hours and 45 minutes, i.e., 2.75 hours.  

f(x) = 10 ceiling(x) + 20  

       = 10 ceiling(2.75) + 20  

       = 10 * 3 + 20  

       = $50  

The cost remains the same at $50. So, I will not save money by returning early.  

Key Concepts Covered  

Exercise

Express each function for the given value.  

1. f(x) = ceiling(x); x = 7.67
2. f(x) = ceiling(x) + 20; x = –2.45
3. f(x) = floor(x); x = –3.4
4. f(x) = floor(x); x = 5.6
5. f(x) = ceiling(x) – 9; x = –10.6
6. f(x) = floor(x) + 8.7; x = –11.2
7. f(x) = floor(x); x = –0.9
8. f(x) = ceiling(x) + 0.89; x = 6.78
9. f(x) = floor(x) – 6.78; x = -35.09
10. f(x) = floor(x) + ceiling(x) – 89; x = –97.6