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How to Find the Range in Math

Find the range in math, a measure of data dispersion. It's the difference between the largest and smallest values. Learn how to calculate it and its importance in data analysis.
8 minutes
How to Find the Range in Math
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All math students have to learn how to find the range at some point during their studies. But you have nothing to fear because the range is one of the easiest topics in mathematics and statistics. It is simply the difference between the highest (maximum) and lowest (minimum) values of a data set.

Mathematicians and statisticians can learn more about how a set of data varies by looking at the range. By the end of this post, you should know how to find the range in math.

Steps to Find the Range in Math

  • To find the range, there has to be a set of numbers, preferably (but not compulsorily) ordered in ascending order, e.g., 1, 2, 3, 4, 5, 6, 7, 8. One advantage of arranging the numbers in ascending order is that you can also quickly calculate other measures, such as the mode, mean, or median of the set.
  • Subtract the least number in the set from the highest. In the above example, the least number is 1, and the highest is 8. Therefore, the range is 8 -1 =7

The Range of a Function

The two main factors of a function are the domain and the range. The domain of a function is the set of all possible input values for that function, while the range of a function is the set of all possible output values for that function. A function is a mathematical relationship that takes as input the values in its domain and returns the values in its range. In a functional relationship, the primary condition is that each input must have just one output.

The Range of a Function Via a Formula

Let f(x) = 3x2 + 6x - 2 be the formula. For this equation (the function of a parabola), one can easily find the value of y if given the value of x.

If it is a quadratic function, then you have to solve for the vertex. This procedure is not needed for other cases, such as a straight line or a function having a polynomial of an odd number [e.g., f(x) = 6x3+2x + 7]. In other words, finding the vertex is only needed if the problem involves a parabola or an equation whose x-coordinate is either squared or raised to an even power. To plot the vertex requires using the formula -b/2a to derive the x-coordinate of the function 3x2 + 6x -2; where a=3, b=6; as shown below:

Therefore, assuming x­= (-b/2a)

Where a=3; b=6

X= -6/2(3)



Solving for y:

fx(-1)=3(-1)2 + 6(-1) -2

3(-1) + (-6)-1) -2

3 – 6 – 2

y= -5

Vertex = (x; y) = (-1; -5)

This vertex (-1, -5) can be plotted on a graph by identifying -1 on the x-coordinate and -5 on the y-coordinate (on the graph’s third quadrant).

Find the function's other points to get a better idea of what it does. However, it should slope upward since it is a parabola with a positive x2 coordinate. Plugging in a few x-coordinates will yield the following corresponding y-coordinates:

  • f(-2) = 3(-2)2 + 6(-2) -2 = -2. A new point on the graph is (-2, -2)
  • f(0) = 3(0)2 + 6(0) -2 = -2. A second point on the graph is (0,-2)
  • f(1) = 3(1)2 + 6(1) -2 = 7. The third point on the graph is (1, 7)

The next step is to locate the range on the graph. The range starts from the lowest point where the graph comes into contact with a y-coordinate. For the above example, this occurs at vertex -5, from where the graph extends infinitely upwards. Hence, the function’s range is:

y = all real numbers ≥ -5.

The Range of a Function Via a Graph

To find the range of a function from a graph requires first identifying both the minimum and maximum values of the function. Assuming the lowest point of the function is -6, and its highest point is 11, then the range of the function is -6 ≤ f(x) ≤ 11.

Note that though -6 is the lowest point of the function, it is still possible that this function could become infinitely smaller and smaller, in which case its set lowest point is infinity. The same rule applies to the function's highest y-coordinate, which is 11. It can also get bigger and bigger forever, in which case its lowest set point is not a real number but infinity.

However, assuming that at point -6, the graph continues an infinite upward trajectory, then the range of the function can be expressed as f(x) ≥ -6. Also, if the graph continues on an infinite downward trajectory at point 11, the range can be expressed as f(x) 11.

The Range of a Function of a Relation

A relation comprises a set of ordered pairs having x and y coordinates. An example is {(4, –5), (2, 4), (5, –1), (7, 2), (2, 4)}. To find the range of a function in a relation, first find out the domain and range of the relation.

Identify all of the y-coordinates of the ordered pairs of the relation. For the example above, the y-coordinates are {-5, 4, -1, 2, 4}.

If any of the coordinate numbers appear twice, then one of them should be deleted from the set. In our example, the number 4 appears twice, so one will have to be removed. Hence, the new set is {-5, 4, -1, 2}.

To obtain the range of the relation, reorder the range data set to start from the smallest number to the largest [that is, in ascending order] to yield -5, -1, 2, 4}.

Check that the relation has the function property. As noted earlier, in a functional relationship, the primary condition is that each input must have just one output.

To put it another way, in order for a relationship to be considered a function, each number associated with an x-coordinate must have ONLY ONE corresponding y-coordinate. For instance, the relation {(3, 4) (3, 5), (6, 2)} is not a function because the number 3 has different corresponding values (4 and 5, respectively).

The Range of a Function in a Word Problem

To find the range of a function as a word problem, you need to read and understand the passage and then transform it into a function.

Assuming the word problem is phrased as: "Each league team will get 3 points for every game it wins," It follows, therefore, that the number of points each team gets is a function of the games won.

Express the word problem in functional form. Hence, let P denote the available points and w the number of games won. The function can then be written as:


So if a team wins six games, then simply multiply 6 by 3 to obtain 18 points.

The next step is to find the domain, which you have to do before you can figure out the range. The domain is made up of all of the possible values for w in the equation. For example, no team can win fractional points, e.g., 3 and a half points.

Also, since the number of teams in the league is not given, let us assume each team can theoretically win an infinite number of points. Most importantly, all the teams in the league can win 3 points, and none can win negative points. Therefore, the function’s domain is w = any non-negative integer.

After finding the domain, it is now easy to determine the range. In this case, the range is the possible number of points each team can get by beating other competing teams in the league.

Since it has already been shown that the equation is P(w)=3w and that the domain must be a non-negative integer, it is easy to find the range or output by plugging in any non-negative integer.

For instance, if Team A wins 7 games, then P(7) = 7 x 3 = 21 points. If it wins 9 games, then P(9) = 9 x 3 = 27 points. Hence, the range of this function includes any non-negative integer that is a multiple of 3. This means that every possible output for this function's input must be a non-negative integer that is a multiple of 3.

Final Thoughts

Among the important attributes of a mathematical or statistical data set are the center of the data and the spread of such data. Several ideas, like the mean, median, mode, and midrange, can be used to find the middle of a set of numbers. Similarly, data spread can also be measured in a variety of ways. One such way is the "range," which is the simplest and crudest measure of data spread.

As you can see above, the range is just one of several ways data can be measured in math. So, after learning how to find the range in math, you should also learn about other math concepts, such as the interquartile range, standard deviation, and more. This is where Learner.com can help.

Learner was started by teachers who believe that students learn best when they are both passionate and good at what they do. We offer the best online tutors, custom-matched through our proprietary screening and vetting process. Tutors need more than subject matter expertise – they must possess the right technical and people skills to inspire and motivate today’s learners.

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About the author:

Mike developed his passion for education as a math instructor at Penn State University. He expanded his educational experience launching and running an Executive Education business - training over 100,000 students per year. As the CEO of Learner, Mike focuses on accelerating learning and unleashing the potential of students. 

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