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

What are Domain and Range?

In basic terms, domain and range refer to different values in in contrast to one another. For instance, let's check out the grading system of a school where a student earns 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 result. Expressed mathematically, 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 can be stated as an instrument that takes respective pieces (the domain) as input and makes specific other pieces (the range) as output. This can be a tool whereby you could get different items for a particular amount of money.

In this piece, we review 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 instance, 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 set 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 take a look at the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can apply any value for x and acquire a corresponding output value. This input set of values is necessary to find the range of the function f(x).

However, there are specific cases under which a function must not be stated. So, 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 group of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we might see that the range is all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

However, as well as with the domain, there are particular terms under which the range may not be defined. 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 explains a batch of numbers applying two numbers that represent the bottom and upper boundaries. For instance, the set of all real numbers in the middle of 0 and 1 could be identified applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be identified using graphs. So, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we must 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 could watch 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,∞).

That’s 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 multiple types of functions. Let's take a look at some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Consequently, the domain for an absolute value function consists of 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. Therefore, each real number can be a possible input value. As the function only 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 alternates 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. For that reason, the domain of the function consists of 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 Examples 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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