What Is Range in Mathematics?

Professor explaining range to student in classroom
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You have two different ways to define range in math. If you're doing statistics, the "range" usually means the difference between the highest and lowest values in a set of data. If you're doing algebra or calculus, the "range" is understood to be the set of possible results, or output values, of a function.

Range in Statistics

If you're asked to find the range in statistics, you're simply being asked to find the highest and lowest values in your data set, and then find the difference between them. Any time you hear "difference," it's a clue that you're about to subtract, so the formula you'll use is:

\text{highest value} - \text{ lowest value} = \text{ range}

Tips

  • Don't forget to include any units (feet, inches, pounds, gallons, etc.) that may be appended to your data set.

Example 1:​ Imagine that you snuck a peek at your teacher's notebook, and you saw that so far, the students' grade percentages in class are {95, 87, 62, 72, 98, 91, 66, 75}. Curly brackets are often used to enclose a set of data, so you know everything inside the curly brackets belongs together.

What's the range of this data set or, to put it another way, the range of the students' grades? First, identify the highest data point (98) and the lowest data point (62). Next, subtract the lowest value from the highest value:

98 - 62 = 36

So the range of this particular data set is 36 percentage points.

The Range of a Function

When you begin studying functions in mathematics, you'll run into a second definition of range. To understand range, it helps to think of functions as little math machines. The set of values you can put into the math machine are called the domain (another very important concept). The set of possible results, once you crank those values through the math machine, is called the ​codomain​. And the set of actual results or outputs you get is called the ​range​.

There are a couple of important relationships between range and domain that you need to understand. First, each value in the domain corresponds to only one value in the range of your function. If any value(s) in the domain correspond to more than one value in the range, you might have a relationship between the two sets of data, but it's not technically classified as a function. However, it is possible for more than one domain value to correspond to the same value in that function's range.

One of the best ways to make sense of this is to imagine your very own math class. The students in the class represent the domain (or the information that goes into the function), while the class itself is the function or "math machine." Your final grades represent the range, or what you get after cranking the elements of the domain (students) through the function (math class).

When you look at that example, you can intuitively see that each student is going to receive only one final grade once class is over. Each value in the domain corresponds to only one value in the range. However, it is possible for more than one student to get the same grade. For example, there might be two or three students in your class that studied very hard and managed to get a 96 percent as their final grade. Multiple values in the domain can correspond to a single value in the range.

Example 2:​ Imagine that you're dealing with the function ​x2, with a domain restricted to { −3, −2, −1, 1, 2, 3, 4}. What's the range of this function?

Although you'll learn more advanced ways of finding the range later on, for now, the simplest way to find the range of this function is to apply the function to each element of the domain, and track your results. In other words, insert each element of the domain, one at a time, as ​x​ in the function ​x2. This gives you a set of results:

\{9, 4, 1, 1, 4, 9, 16\}

But as you can see, some elements are repeated there. Recalling the example of math grades as a function, that's okay; more than one student can end up with the same grade, or more than one element of the domain can "point" to the same element in the range. But you don't want to write down the repeated elements when you give the range. So, your answer is simply:

\{1, 4, 9, 16\}

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