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

What are Domain and Range?

To put it simply, domain and range apply to different values in comparison to each other. For example, let's take a look at 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 math, the result 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 might be defined as a tool that takes respective objects (the domain) as input and produces certain other items (the range) as output. This could be a machine whereby you can get multiple snacks for a particular quantity of money.

In this piece, we will teach you the fundamentals 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 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 set of all input values for the function. To clarify, 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 cloud plug in any value for x and obtain itsl output value. This input set of values is necessary to discover the range of the function f(x).

But, there are certain cases under which a function cannot be specified. For example, if a function is not continuous at a particular point, then it is not specified for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. To be specific, it is the batch of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.

Nevertheless, as well as with the domain, there are particular conditions under which the range cannot be stated. For example, if a function is not continuous at a certain point, then it is not defined for that point.

Domain and Range in Intervals

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

(0,1)

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

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

(-∞,∞)

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

The range of this function can be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be identified with 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 chart these points on a coordinate plane, it will look like this:

As we can see from the graph, the function is stated for all real numbers. This means that the domain of the function is (-∞,∞).

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

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

How do you find the Domain and Range?

The process 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 structure y=|ax+b| is defined for real numbers. Consequently, the domain for an absolute value function contains 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. For that reason, any real number can be a possible input value. As the function just 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 shifts between -1 and 1. In addition, the function is stated for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Just 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 just for x ≥ -b/a. Therefore, the domain of the function includes 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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