In this lesson, we’ll examine the greatest integer function. We’ll define this function through words and examples, and lastly, we’ll take a look at different real world scenarios when this type of function can be used.

## Greatest Integer Function

Have you ever mailed a package? If so, you probably noticed that the shipping cost depends on the weight of the package. For instance, suppose the post office has its priority shipping rates listed as shown like this:

Weight Cost
1 lb. up to 2 lbs.

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\$1
2 lbs. up to 3 lbs. \$2
3 lbs.

up to 4 lbs.

\$3
4 lbs. up to 5 lbs. \$4
5 lbs. up to 6 lbs. \$5
6 lbs.

up to 7 lbs.

\$6
7 lbs. up to 8 lbs. \$7
8 lbs. up to 9 lbs. \$8
9 lbs. up to 10 lbs.

\$9

The cost depends on the package weight, so the cost is a function of the package weight. However, here’s something interesting that’s a little different from the functions we’re used to seeing. Did you notice that the shipping cost for any package that has weight between x pounds and x + 1 pounds is the same? For example, if you had a package that weighed 1.2 pounds, it would cost \$1 to ship, and if you had a package that weighed 1.8 pounds, it would also cost \$1.

In mathematics, this is a special type of function called the greatest integer function. It is also sometimes called a step function. There are different notations that we can use for the greatest integer function, including these:

The greatest integer function is a function such that the output is the greatest integer that is less than or equal to the input.

That makes sense, but it’s a little confusing. To clarify, let’s consider it another way. The greatest integer function takes an input, and the output is given based on the following two rules:

1. If the input is an integer, then the output is that integer
2. If the input is not an integer, then the output is equal to the next smallest integer

We can also think of it as taking an input and rounding down to the nearest integer. That makes things a bit more clear.

So if we are given a number, say 18, and we plug it into the greatest integer function, we would get 18 back out since 18 is an integer. On the other hand, if we were given a non-integer number, say 5.27, the output would be equal to the next smallest integer, which is 5.

I think we’ve got it!To solidify our understanding of this concept, let’s consider some more examples:

## Graphing the Greatest Integer Function

Alright, we’ve got this greatest integer function concept down, but this may leave you curious about what a graph of this type of function might look like. Well, I’ll give you a hint – they don’t call it a step function for nothing! Let’s take a look at the graph of our postage example:

Since the shipping cost is the same for a package weighing x pounds up to x + 1 pounds, the graph ends up being a series of horizontal lines that look like a set of steps – ah-ha, a step function! If we extend the graph in both directions using this same rule, the resulting graph is the graph of the greatest integer function.We see that the domain (or the set of inputs) of the greatest integer function consists of all real numbers. However, the range (or the set of outputs) only consists of the values of the ‘steps.’ That is, the range of the greatest integer function consists of the integers, since the outputs of this function will always be an integer.

## Another Application

Uses for the greatest integer function can be found quite often in the world around us. – especially when it comes to billing and cost analysis. For instance, utility bills, such as gas, phone, or electric bills, are often charged based on increments of the amount of gas, minutes, or electricity used.For instance, suppose that your phone company charges \$1.

00 per minute for long distance. Therefore, if you talk for 1 minute, you get charged \$1.00.

However, if you talk for 1.5 minutes or any fraction of a minute over 1, you still get charged \$1.00, until you get to 2 minutes, then you bump up to \$2.

00. This is exactly the greatest integer function! This tells us that we can model the long distance charge by inputting the number of minutes talked into the greatest integer function.To illustrate this, suppose your long distance call lasts 7.98 minutes.

How much will you be charged? We plug 7.98 into the greatest integer function, and get 7 back out, because 7 is the next greatest integer that is less than 7.98. Well, that’s pretty handy, wouldn’t you agree?

## Lesson Summary

The greatest integer function is a function that takes an input and gives an output that is the greatest integer that is less than or equal to the input. This type of function is also called a step function because its graphs look exactly like a set of steps, with domain equal to all real numbers and range equal to the integers.

There are a number of different notations we can use to represent the greatest integer function:

Using the greatest integer function is actually quite easy, once we get used to the concept of rounding down to the nearest integer. This is good to know, because as we’ve seen, this function can definitely show up in our daily lives!