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

What are Domain and Range?

In simple terms, domain and range apply to different values in comparison to each other. For instance, let's check out grade point averages of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the average grade. Expressed mathematically, the result is the domain or the input, and the grade is the range or the output.

Domain and range can also be thought of as input and output values. For example, a function can be defined as a machine that takes particular objects (the domain) as input and produces certain other objects (the range) as output. This can be a instrument whereby you can obtain different treats for a particular amount of money.

Here, we review the essentials 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 group of all input values for the function. To put it simply, it is the batch 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 might plug in any value for x and acquire a corresponding output value. This input set of values is needed to figure out the range of the function f(x).

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

The Range of a Function

The range of a function is the group of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we could see that the range will be all real numbers greater than or the same as 1. Regardless of the value we plug in for 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 particular point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range can also be identified via interval notation. Interval notation expresses a set of numbers working with two numbers that identify the lower and higher boundaries. For example, the set of all real numbers in the middle of 0 and 1 can be classified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be identified via graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we need to 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 defined 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 creates all real numbers greater than or equal to 1.

How do you figure out the Domain and Range?

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

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified 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. Consequently, 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. Further, 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 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 includes 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 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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