## 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. **

**What function can you use to represent this situation?**

**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. **

**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.

**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. **

**f(x) = ceiling(x); x = 2.65**

**f(x) = floor(x); x = 2.19**

**Solution:**

- f(x) = ceiling(2.65) = 3

- 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.

- f(x) = ceiling(x); x = 7.67
- f(x) = ceiling(x) + 20; x = –2.45
- f(x) = floor(x); x = –3.4
- f(x) = floor(x); x = 5.6
- f(x) = ceiling(x) – 9; x = –10.6
- f(x) = floor(x) + 8.7; x = –11.2
- f(x) = floor(x); x = –0.9
- f(x) = ceiling(x) + 0.89; x = 6.78
- f(x) = floor(x) – 6.78; x = -35.09
- f(x) = floor(x) + ceiling(x) – 89; x = –97.6

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: