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 example, let's take a look at the grade point calculation 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. In math, the score 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 could be defined as a machine that takes respective items (the domain) as input and produces certain other items (the range) as output. This can be a machine whereby you might obtain several items for a specified amount of money.

Here, we discuss 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 indicate the x-values and y-values. For example, 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, 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 batch 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 review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can plug in any value for x and acquire itsl output value. This input set of values is necessary to figure out the range of the function f(x).

Nevertheless, there are particular conditions under which a function must not 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 group of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range would be all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

But, just like with the domain, there are specific conditions under which the range cannot be specified. For instance, 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 can also be identified using interval notation. Interval notation indicates a set of numbers applying two numbers that identify the bottom and upper limits. For example, the set of all real numbers in the middle of 0 and 1 might be identified applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we need 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 plot these points on a coordinate plane, it will look like this:

As we could watch from the graph, the function is specified for all real numbers. This shows us 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 determine the Domain and Range?

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

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Therefore, 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. Consequently, any real number might be a possible input value. As the function only returns positive values, the output of the function includes 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 stated 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 form y= √(ax+b) is specified 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 always 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 Examples 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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