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High School Algebra II: Tutoring Solution26 chapters | 274 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll learn what a step function is and become comfortable with this type of function. We'll examine common examples, such as the floor and ceiling functions and the heaviside function.

Mathematically speaking, a **step function** is a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in-between. For this reason, it is also sometimes called a staircase function.

A step function has a constant value on given intervals, but the constant is different for each interval. The constant value on each interval creates the series of horizontal lines, and the fact that the constant is different for each interval creates the jumps in between each horizontal line segment. This is why the graph of a step function looks like a set of stairs.

To help us better understand this definition, let's consider a real-world example. Suppose I start a cleaning business. I decide to charge each client based on how many hours I work for that client.

My charge structure is as follows:

- Less than one hour: $10.00
- One hour up to two hours: $20.00
- Two hours up to three hours: $30.00
- Three hours up to four hours: $40.00

This means that if I work anywhere from 1-59 minutes, I charge $10.00. If I work anywhere from 1 hour to 1 hour and 59 minutes, I charge $20.00, and so on. We see that the amount I charge remains constant on each hour-long interval, but varies on each of those intervals, because it goes up each hour. Here is the graph corresponding to this example.

It is easy to see from this graph why a step function is sometimes called a staircase function; it looks exactly like a set of stairs.

In mathematics, a common example used to introduce step functions is the **greatest integer function** (also called the floor function). The greatest integer function is often represented as *x* with bottom brackets around it.

It maps each real number *x* to the greatest integer that is less than or equal to *x*. Whatever we put into the greatest integer function, we get the greatest integer that is less than or equal to that input as our output. Here is the graph of the greatest integer function.

It is easy to see that the greatest integer function is a step function from its graph.

Let's consider another example of a step function that is very similar to the greatest integer function. It is called the **least integer function** (also known as the ceiling function). The least integer function is often represented as *x* with top brackets around it.

The least integer function is a step function that assigns, or maps, each real number *x* to the smallest integer that is greater than or equal to *x*. Whatever we put into the least integer function, we get the smallest integer that is less than or equal to that input as our output. The least integer function is shown in the graph on screen.

We see that the graph of the least integer function looks like a set of stairs, as it should since this is a step function.

Notice, the floor function contains the left endpoint of each interval and not the right, but the ceiling function contains the right endpoint of each interval and not the left; this is where they get their names from. The ceiling function contains the greater endpoint of each interval (the ceiling), and the floor function contains the smaller endpoint of each interval (the floor). Our cleaning business example is a variation of the floor function because it contains the smaller endpoint of each interval.

Another very simple example of a step function is the **heaviside step function**, which is also known as the unit step function and is used in engineering and in the study of electric currents. This function, denoted by *H*, is a function defined as follows:

- If
*x*< 0, then*H(x)*= 0 - If
*x*= 0, then*H(x)*= 1/2 - If
*x*> 0, then*H(x)*= 1

For example, *H*(4.5) = 1, *H*(-2.35) = 0, and *H*(0) = 1/2. Thus, the Heaviside function has just one step, as shown in its graph, but it still satisfies the definition of a step function.

In mathematics, the **step function** is a function that has a constant value along given intervals, with the constant value varying between intervals. The name of this function comes from the fact that when you graph the function, it looks like a set of steps or stairs.

We explored three classic examples of step functions. The **greatest integer function** is defined as the greatest integer that is less than or equal to our input. The **least integer function** is defined as the least integer that is greater than or equal to our input. The **heaviside function** is equal to 0 when our input is less than 0, equal to 1/2 when our input is equal to 0, and equal to 1 when our input is greater than 0.

**Step function** - a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in-between

**Greatest integer function** - also known as the floor function; this function is a common example used to introduce step functions

**Least integer function** - also known as the ceiling function; this is a step function that assigns, or maps, each real number *x* to the smallest integer that is greater than or equal to *x*

**Heaviside step function** - the unit step function; used in engineering and in the study of electric currents

Subsequent to viewing the lesson on step functions, confirm what you learned by doing the following:

- Point out step functions on a graph
- Identify and define three different types of step functions

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High School Algebra II: Tutoring Solution26 chapters | 274 lessons

- What is the Correct Setup to Solve Math Problems?: Writing Arithmetic Expressions 5:50
- Understanding and Evaluating Math Formulas 7:08
- Expressing Relationships as Algebraic Expressions 5:12
- Evaluating Simple Algebraic Expressions 7:27
- Combining Like Terms in Algebraic Expressions 7:04
- Practice Simplifying Algebraic Expressions 8:27
- Negative Signs and Simplifying Algebraic Expressions 9:38
- Writing Equations with Inequalities: Open Sentences and True/False Statements 4:22
- Common Algebraic Equations: Linear, Quadratic, Polynomial, and More 7:28
- Defining, Translating, & Solving One-Step Equations 6:15
- Solving Equations Using the Addition Principle 5:20
- Solving Equations Using the Multiplication Principle 4:03
- Solving Equations Using Both Addition and Multiplication Principles 6:21
- Collecting Like Terms On One Side of an Equation 6:28
- Solving Equations Containing Parentheses 6:50
- Translating Words to Algebraic Expressions 6:31
- How to Solve One-Step Algebra Equations in Word Problems 5:05
- How to Solve Equations with Multiple Steps 5:44
- How to Solve Multi-Step Algebra Equations in Word Problems 6:16
- Additive Identity: Definition & Examples 4:56
- Step Function: Definition, Equation & Examples 4:44
- Go to Algebra II - Algebraic Expressions and Equations Review: Tutoring Solution

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