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

What are Domain and Range?

In basic terms, domain and range coorespond with multiple values in comparison to each other. For instance, let's check out grade point averages of a school where a student receives an A grade for an average between 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 score 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 instance, a function could be stated as a tool that catches respective items (the domain) as input and makes particular other pieces (the range) as output. This might be a instrument whereby you can buy different treats for a particular amount of money.

In this piece, we discuss the fundamentals 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 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, for 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 instance, 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 acquire a corresponding output value. This input set of values is required to figure out the range of the function f(x).

But, there are specific cases under which a function cannot be defined. For example, 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 set of all possible output values for the function. To put it simply, it is the set of all y-coordinates or dependent variables. So, working with 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.

But, just as with the domain, there are specific terms under which the range must not be specified. For example, 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 represented via interval notation. Interval notation indicates a batch of numbers working with two numbers that represent the bottom and upper bounds. For example, the set of all real numbers between 0 and 1 might be identified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified with graphs. For example, let's review the graph of the function y = 2x + 1. Before creating 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 chart these points on a coordinate plane, it will look like this:

As we can look from the graph, the function is specified 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 produces all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The task of finding domain and range values differs for multiple types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is specified for real numbers. For that reason, 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 as y = ax, where a is greater than 0 and not equal to 1. Therefore, each real number could 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 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]

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. Consequently, the domain of the function contains 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 consists of 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

Let Grade Potential Help You Learn Functions

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