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

What are Domain and Range?

In simple terms, domain and range refer to different values in in contrast to each other. For instance, let's take a look at the grade point calculation of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the total score. Expressed mathematically, the result 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 instance, a function might be stated as a machine that takes respective items (the domain) as input and produces particular other objects (the range) as output. This could be a machine whereby you might get different items for a particular amount of money.

Here, we review the fundamentals of the domain and the range of mathematical functions.

What are the Domain and Range of a Function?

In algebra, the domain and the range indicate 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, because 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 group of all x-coordinates or independent variables. For instance, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can apply any value for x and acquire a corresponding output value. This input set of values is needed to find the range of the function f(x).

Nevertheless, there are specific terms under which a function may not be defined. 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 batch of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range is all real numbers greater than or equivalent tp 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

However, as well as with the domain, there are particular terms under which the range must not be stated. For example, if a function is not continuous at a particular point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range might also be classified via interval notation. Interval notation expresses a group of numbers using two numbers that classify the bottom and higher boundaries. For instance, the set of all real numbers between 0 and 1 might be identified working with interval notation as follows:

(0,1)

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

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

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be represented with graphs. For example, let's consider 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 chart these points on a coordinate plane, it will look like this:

As we can see from the graph, the function is defined 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 produces 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 is different for various 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. 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. Therefore, every real number can be a possible input value. As the function only 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 oscillates among -1 and 1. Also, the function is defined 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 stated only for x ≥ -b/a. Consequently, the domain of the function consists of 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 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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