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 example, let's consider grade point averages of a school where a student gets an A grade for a cumulative score of 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the result. In mathematical terms, 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 instance, a function might be stated as a machine that catches respective objects (the domain) as input and generates certain other pieces (the range) as output. This can be a machine whereby you can get different items for a specified quantity 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 refer to the x-values and y-values. For example, 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. In other words, it is the group of all x-coordinates or independent variables. For example, let's consider the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we cloud apply any value for x and acquire itsl output value. This input set of values is required to discover the range of the function f(x).

But, there are particular cases under which a function may not be defined. For instance, if a function is not continuous at a specific 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 put it simply, it is the batch of all y-coordinates or dependent variables. For example, using 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 apply to x, the output y will always be greater than or equal to 1.

However, just like with the domain, there are specific conditions under which the range must not be stated. For example, if a function is not continuous at a particular point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range might also be classified using interval notation. Interval notation indicates a set of numbers applying two numbers that identify the bottom and upper limits. For instance, the set of all real numbers between 0 and 1 might be classified applying interval notation as follows:

(0,1)

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

Equally, the domain and range of a function could be classified via interval notation. So, let's look at 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 identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be identified using graphs. For example, let's consider the graph of the function y = 2x + 1. Before creating a graph, we must discover 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 graph these points on a coordinate plane, it will look like this:

As we might look 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 produces 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 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 defined for real numbers. Consequently, 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. For that reason, every real number can be a possible input value. As the function just returns 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 oscillates among -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 defined just 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 includes 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

Discover 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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