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 in contrast to each other. For instance, let's take a look at grade point averages of a school where a student receives 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 might also be thought of as input and output values. For example, a function can be defined as a tool that catches specific objects (the domain) as input and produces particular other pieces (the range) as output. This can be a instrument whereby you can buy several snacks for a particular amount of money.

Here, we discuss 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 indicate the x-values and y-values. For example, let's view 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. To put it simply, it is the batch of all x-coordinates or independent variables. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and get a corresponding output value. This input set of values is necessary to discover the range of the function f(x).

However, there are particular terms under which a function cannot be specified. For example, if a function is not continuous at a certain point, then it is not specified 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 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 the same as 1. Regardless of the value we apply to 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 specified. For example, if a function is not continuous at a specific point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range can also be represented via interval notation. Interval notation expresses a set of numbers using two numbers that classify the bottom and upper bounds. For example, the set of all real numbers among 0 and 1 might be represented using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be classified with graphs. So, let's review the graph of the function y = 2x + 1. Before plotting a graph, we must 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 might look 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 because 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 differs for various types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is defined 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 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 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 among -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]

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 structure y= √(ax+b) is specified 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 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 Questions on Domain and Range

Realize 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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