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

What are Domain and Range?

In simple terms, domain and range apply to multiple values in comparison to one another. For instance, let's check out the grade point calculation of a school where a student earns an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the total score. In mathematical terms, the score is the domain or the input, and the grade is the range or the output.

Domain and range could also be thought of as input and output values. For example, a function could be stated as an instrument that catches specific items (the domain) as input and produces specific other pieces (the range) as output. This could be a instrument whereby you can obtain different items for a particular quantity of money.

In this piece, we will teach you the basics of the domain and the range of mathematical functions.

What is 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, whereas 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 batch of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud plug in any value for x and get itsl output value. This input set of values is required to find the range of the function f(x).

However, there are particular cases under which a function must not be defined. For example, if a function is not continuous at a particular 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 group of all y-coordinates or dependent variables. For instance, 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 plug in for x, the output y will always be greater than or equal to 1.

However, just like with the domain, there are specific terms under which the range must not be defined. 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 could also be identified using interval notation. Interval notation indicates a batch of numbers using two numbers that identify the bottom and higher bounds. For example, the set of all real numbers in the middle of 0 and 1 might be identified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function could be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented using graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must determine 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 might see from the graph, the function is stated for all real numbers. This tells us that the domain of the function is (-∞,∞).

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

That’s because 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 differs for various types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is specified for real numbers. Consequently, 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. Therefore, any real number can be a possible input value. As the function just produces 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 varies among -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]

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 structure 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 will always 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 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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