How to Find the Range of a Function – Methods & Practice
The Range
In mathematics, a range is a difference between the lowest and highest values of a numeral. In {7, 15, 4, 6, 9} the lowest value is 4, and the highest is 15, thus the range is 15 − 4 = 11. The range can also imply all the values of the output of a function. Moreover, when you start studying functions in mathematics, you'll encounter a second definition of range. To better understand range, it aids to think of functions as tiny math machines.
Range of a Function
Talking about the range of a function definition, it is the set of outputs the function accomplishes when it is pertained to its whole set of outputs. In the function machine metaphor, the range is the set of items that arise out of the machine when you insert in all the inputs.
For instance, when we apply the function notation f: R→R, we imply that f is a function →from the real numbers →to the real numbers. By this notation, we are aware that the domain (set of all inputs) of ‘f’ is the set of all possible inputs (the codomain) and as well the set of all real numbers.
But, without having to know the function f, we will be unable to identify what its outputs are further cannot even determine what its range is. All we know is that the range should be a subset of the codomain, so the range should be a subset (likely to be the whole set) of the real numbers. It is possible objects are available in the subset of codomain for which there are no inputs and for which the function will output that object.
For instance, we could describe a function f: R→R as f(x) =x2. Seeing that f(x) will invariably be non-negative, the number −3 is in the codomain set of f, but it is not in the range, since there is no input of x for which f(x) =−3. For this f, the codomain is the set of all real numbers whereas the range is the set of non-negative real numbers.
Domain and Codomain in Range
The set of values we can insert into the math machine are known as the domain (another very important concept in the range). The set of possible outcomes, once we crank those values via the math machine, is known as the co domain. And the set of actual outputs or outcomes we obtain is called the range.
Interquartile Range
The Interquartile Range also known as IQR, defines the mid ( 50%) of values when arranged from lowest to greatest in the data set. In order to determine the interquartile range (IQR), we need to first find the median (middle value) of the lower and upper half of the set of data. These values are assigned as quartile 1 (Q1) and quartile 3 (Q3). The IQR is thus the difference between Q3 and Q1.
Solved Examples
Example:
Think that you happen to view your math’s teacher's notebook, and you snuck the peek so far that you saw the students' grade percentages in class are {91, 84, 37, 53, 52, 88, 46, 62}. Now, you need to find out the range of this data set or we can say the range of the students' grades?
Solution:
First, we need to determine the highest as well as the lowest value of the data set i.e
The highest data point = 91
The lowest data point= 37
Next, subtract the lowest value from the highest value determined:
91 - 37 = 54
Thus, the range of this specific data set is 54 percentage points.
Example:
Mr Alex drove through 8 southern states on his summer vacation. Fuel prices varied from state to state he travelled. Calculate the range of fuel prices?
Rs. 2.79, Rs. 0.61, Rs. 2.96, Rs. 3.09, Rs. 1.64, Rs. 2.25, Rs. 3.73, Rs. 1.67
Solution:
Arranging the data from least to greatest, we obtain,
0.61, 1.64, 1.67, 2.25, 2.96, 2.79, 3.09, 3.73
highest - lowest = 3.73 – 0.61 = ＄0.48
Answer: The range of fuel prices is Rs. 3.12
Fun Facts
While finding the range, curly brackets are commonly used to enclose a set of data, so you are aware everything inside the curly brackets belongs together.
FAQs on Range in Maths: Explained with Examples
1. What does the term 'range' signify in mathematics?
In mathematics, the term 'range' has two primary meanings depending on the context:
In Statistics: The range is the simplest measure of spread or dispersion in a dataset. It is calculated as the difference between the highest and lowest values. For instance, in the dataset {4, 6, 9, 3, 7}, the highest value is 9 and the lowest is 3, so the range is 9 - 3 = 6.
In Functions and Relations: The range is the set of all possible output values (y-values) that a function can produce from its corresponding set of input values (the domain). For the function f(x) = x², the range is all non-negative real numbers, as the output is always zero or positive.
2. How do you calculate the range for a given set of data?
To calculate the range of a statistical dataset, you follow a simple two-step process. The formula is: Range = Maximum Value - Minimum Value.
Step 1: Arrange the data to identify the highest (maximum) and lowest (minimum) values in the set.
Step 2: Subtract the minimum value from the maximum value.
For example, to find the range of the scores {88, 72, 95, 79, 84}, the maximum value is 95 and the minimum is 72. Therefore, the range is 95 – 72 = 23.
3. What is the difference between the domain and range of a function?
The primary difference between domain and range lies in their roles within a function. Think of a function as a machine:
The Domain is the set of all permissible inputs for the function. These are the values you are allowed to put into the 'machine'. In a coordinate pair (x, y), these are the x-values.
The Range is the set of all possible outputs the function generates after processing the inputs. These are the values that come out of the 'machine'. In a coordinate pair (x, y), these are the y-values.
For the function y = x + 5, the domain could be all real numbers, and the corresponding range would also be all real numbers.
4. Does an outlier (an extremely high or low value) affect the range of a dataset? Explain how.
Yes, an outlier has a significant impact on the range. Because the range is calculated using only the absolute highest and lowest values in a dataset, a single extreme value can drastically alter it. For example, consider the dataset {10, 12, 15, 18}. The range is 18 - 10 = 8. If we add an outlier, say 100, the new dataset is {10, 12, 15, 18, 100}. The new range becomes 100 - 10 = 90. This shows that the range is a sensitive measure of spread and may not always represent the typical spread of the data if outliers are present.
5. In the context of a function's graph, do the x-values or y-values represent the range?
The range of a function is represented by the y-values. A common point of confusion is mixing up the axes. The set of all possible x-values on the graph corresponds to the domain, while the set of all possible y-values that the graph covers vertically represents the range. So, to find the range from a graph, you should look at the vertical spread of the curve.
6. Why is understanding the range important in real-world data analysis?
Understanding the range is crucial in real-world scenarios because it provides a quick and simple snapshot of the variability or spread in a set of data. For example:
In Weather Forecasting: The daily temperature range (difference between the day's high and low) helps in understanding climate fluctuations.
In Finance: The range of a stock's price over a day shows its volatility.
In Quality Control: The range of measurements for a product can indicate the consistency of the manufacturing process.
It helps in identifying the extent of variation and detecting potential outliers instantly.
7. How does the 'range' differ from other statistical measures like 'mean' and 'median'?
The key difference is what they describe about a dataset. The range is a measure of spread or dispersion, telling you how spread out the data is from the lowest to the highest value. In contrast, the mean (average) and median (middle value) are measures of central tendency, which describe the center or typical value of the data. Essentially, the range describes the width of the data, while the mean and median describe its center.
8. Can the range of a function ever be just a single number? Provide an example.
Yes, the range of a function can be a single number. This occurs in a constant function, where the output value is the same for every input. For example, consider the function f(x) = 7. No matter what value of 'x' you choose as an input (e.g., -5, 0, 100), the output 'y' will always be 7. Therefore, the set of all possible output values, which is the range, consists of only one number: {7}.
