Step Function

A step function (also known as a staircase function) is a piecewise-defined function that is defined by two or more equations. Each piece of a step function is a point or horizontal line segment. Together, they create the appearance of steps on a graph.

Basic form of the step function

The equation for the basic form of a step function is $f\left(x\right)=\lfloor x\rfloor$ . In many cases, the brackets around the x represent the greatest integer function or floor function. The greatest integer function is a function that results in the integer part of the given real number. For every value of x, $f\left(x\right)$ takes the value of the greatest integer that is less than or equal to x.

To find the value of the greatest integer, you'll want to round down to the nearest integer. For example, for $\left[3.2\right]$ , look for the greatest integer that is less than 3.2. In this case, the greatest integer that is less than 3.2 is 3.

Here are a few more examples:

$\lfloor 5.7\rfloor =5$

$\lfloor 1.9\rfloor =1$

$\lfloor 2.4\rfloor =2$

What if you want to find the greatest integer of a negative number? Remember, numbers on the negative side of the number line get smaller as you move left. So, the value of the greatest integer will look like this:

$\lfloor -6.1\rfloor =-7$

$\lfloor -1.3\rfloor =-2$

$\lfloor -7.8\rfloor =-8$

$\lfloor -3.4\rfloor =-4$

It's also good to note that if you encounter integers, the greatest integer will be the number itself:

$\lfloor 2\rfloor =2$

$\lfloor 1\rfloor =1$

$\lfloor 0\rfloor =0$

$\lfloor -1\rfloor =-1$

Domain and range of the step function

With step functions, it's common to see that for each input value of x, the output value of $f\left(x\right)$ or y takes the value of the greatest integer, less or equal to the value of x. A good way to see the input and output values of step functions is to create an input-output table.

For $y=\lfloor x\rfloor$

$x$	0	0.25	0.50	0.75	1	1.25	1.50	1.75	2	2.25	2.5	2.75
$y=\left[x\right]$	0	0	0	0	1	1	1	1	2	2	2	2

The value of x can be any real number. It's for this reason that the domain of the step function is commonly $ℝ$ . On the other hand, the range (all of the y-values or outputs) of the step function is integers, represented by $\mathbb{Z}$ .

Graphs of step functions

Now that we've looked at some basics of the step function, let's explore graphed step functions.

Example 1

In the following example, y is already defined, and we're identifying x values. The function is defined for all the integers as:

$y=-2forx<1$

$y=3forx\ge 1$

The first equation shows that all y-values equal -2 while all x-values are integers less than 1 (0, -1, -2, etc.). The second equation shows that all y-values equal 3 while all x-values are integers equal to and greater than 1 (1, 2, 3, etc.).

Here's the resulting graph:

This function represents an infinite number of discrete points.

Example 2

Here is a graph of the function $y=-n$ for n $<x\le n+1$ where $x\in ℝ$ , $n\in \mathbb{Z}$ (x is a member of the set of real numbers, and n is a member of the set of integers):

Since the value of y is -n, you have an infinite number of y-values (integers) each accompanied by a set of x-values (real numbers). This produces an infinite number of horizontal line segments.

Also, note that each line segment includes the right endpoint but not the left endpoint. In this case, the right endpoint (closed dot) represents a point on the graph and the lack of a left endpoint (open dot) means there is no point at that number on the line segment. In other words, (2, -2), (1.9, -2), (1.8, -2), (1.7,-2), etc. are points on the graph, but (1, -2) are not. Instead, the x-value 1 jumps up to the next line segment.

Practice questions on the step function

a. What does the symbol $ℝ$ represent?

Real numbers

b. What is the greatest integer of $\left[6.3\right]$ ?

6

c. What does the symbol ℤ represent?

Integers

d. What is the greatest integer of $\left[3.1\right]$ ?

-4

e. What does $x\in ℝ$ mean?

x is a member or exists in the set of real numbers

Topics related to the Step Function

Piecewise-Defined Function

One to One Functions

Input-Output Tables

Flashcards covering the Step Function

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Step Function

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

Get help learning about the step function

Step functions can be quite difficult to grasp for first-time learners. It's not always easy to comprehend the greatest integer function or graphs of step functions. If your student is having a difficult time with step functions or would like to go over them as they prepare for a test or work on a homework assignment, studying alongside a skilled tutor can help. Tutors can address your student's questions and give step-by-step guidance to provide greater clarity.

Tutoring isn't just for struggling students, either. Learners who are eager to move forward with more advanced math concepts can also benefit from tutoring! Learn more about the benefits of tutoring and get signed up by reaching out to the Educational Directors at Varsity Tutors.