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

What are Domain and Range?

In basic terms, domain and range apply to different values in in contrast to each other. For example, let's check out the grade point calculation of a school where a student receives an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the average grade. In math, the total is the domain or the input, and the grade is the range or the output.

Domain and range might also be thought of as input and output values. For example, a function might be specified as a machine that catches particular objects (the domain) as input and makes certain other items (the range) as output. This could be a instrument whereby you might obtain multiple treats for a respective quantity of money.

Here, 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. For instance, let's look at 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 set of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can plug in any value for x and acquire a respective output value. This input set of values is required to discover the range of the function f(x).

Nevertheless, there are specific cases under which a function cannot be defined. So, if a function is not continuous at a specific point, then it is not stated for that point.

The Range of a Function

The range of a function is the set of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range would be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.

But, just like with the domain, there are certain terms under which the range must not be stated. For example, if a function is not continuous at a specific 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 explains a set of numbers working with two numbers that represent the lower and upper boundaries. For example, the set of all real numbers between 0 and 1 might be represented using interval notation as follows:

(0,1)

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

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

(-∞,∞)

This tells us that the function is defined for all real numbers.

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be classified via graphs. So, let's review the graph of the function y = 2x + 1. Before plotting a graph, we have to find 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 see from the graph, the function is defined for all real numbers. This means that the domain of the function is (-∞,∞).

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

This is because the function generates all real numbers greater than or equal to 1.

How do you determine the Domain and Range?

The task of finding domain and range values is different for multiple 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 includes 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Consequently, every real number might be a possible input value. As the function only delivers positive values, the output of the function consists of 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 alternates between -1 and 1. Also, the function is specified for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Take a look at 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 stated only for x ≥ -b/a. Therefore, the domain of the function includes all real numbers greater than or equal to b/a. A square function will consistently result in a non-negative value. So, the range of the function contains all non-negative real numbers.

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

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Examples 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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