Domain and Range - Examples | Domain and Range of a Function

What are Domain and Range?

To put it simply, domain and range refer to different values in in contrast to one another. For instance, let's take a look at the grading system of a school where a student receives an A grade for a cumulative score of 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the average grade. In math, the result is the domain or the input, and the grade is the range or the output.

Domain and range can also be thought of as input and output values. For example, a function might be stated as a tool that takes respective objects (the domain) as input and makes specific other pieces (the range) as output. This can be a tool whereby you might get different snacks for a specified amount of money.

Today, we will teach you the basics of the domain and the range of mathematical functions.

What are the Domain and Range of a Function?

In algebra, the domain and the range refer to the x-values and y-values. So, let's check the coordinates for the function f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8).

Here the domain values are all the x coordinates, i.e., 1, 2, 3, and 4, because the range values are all the y coordinates, i.e., 2, 4, 6, and 8.

The Domain of a Function

The domain of a function is a group of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. For example, let's consider the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can plug in any value for x and get a respective output value. This input set of values is necessary to discover the range of the function f(x).

But, there are particular conditions under which a function must not be defined. So, if a function is not continuous at a specific point, then it is not defined for that point.

The Range of a Function

The range of a function is the group of all possible output values for the function. To be specific, it is the batch of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equal to 1. Regardless of the value we assign to x, the output y will continue to be greater than or equal to 1.

But, just like with the domain, there are specific conditions under which the range may not be defined. For instance, if a function is not continuous at a particular point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range can also be classified via interval notation. Interval notation indicates a set of numbers applying two numbers that identify the lower and upper boundaries. For example, the set of all real numbers between 0 and 1 can be identified using interval notation as follows:

(0,1)

This means that all real numbers more than 0 and less than 1 are included in this group.

Similarly, the domain and range of a function can be classified with interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:

(-∞,∞)

This means that the function is stated for all real numbers.

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented via graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we need to discover all the domain values for the x-axis and range values for the y-axis.

Here are the coordinates: (0, 1), (1, 3), (2, 5), (3, 7). Once we plot these points on a coordinate plane, it will look like this:

As we can watch from the graph, the function is specified for all real numbers. This shows us that the domain of the function is (-∞,∞).

The range of the function is also (1,∞).

This is due to the fact that the function produces all real numbers greater than or equal to 1.

How do you determine the Domain and Range?

The process of finding domain and range values is different for different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Therefore, the domain for an absolute value function consists of all real numbers. As the absolute value of a number is non-negative, the range of an absolute value function is y ∈ R | y ≥ 0.

The domain and range for an absolute value function are following:

• Domain: R

• Range: [0, ∞)

For Exponential Functions

An exponential function is written as y = ax, where a is greater than 0 and not equal to 1. For that reason, any real number might be a possible input value. As the function just returns positive values, the output of the function contains all positive real numbers.

The domain and range of exponential functions are following:

• Domain = R

• Range = (0, ∞)

For Trigonometric Functions

For sine and cosine functions, the value of the function oscillates between -1 and 1. Further, the function is defined for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Just see the table below for the domain and range values for all trigonometric functions:

For Square Root Functions

A square root function in the form y= √(ax+b) is specified just for x ≥ -b/a. Consequently, the domain of the function consists of all real numbers greater than or equal to b/a. A square function always result in a non-negative value. So, the range of the function consists of all non-negative real numbers.

The domain and range of square root functions are as follows:

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Questions on Domain and Range

Find the domain and range for the following functions:

1. y = -4x + 3

2. y = √(x+4)

3. y = |5x|

4. y= 2- √(-3x+2)

5. y = 48

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