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 comparison to one another. For instance, let's take a look at the grade point calculation of a school where a student earns an A grade for a cumulative score of 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 could also be thought of as input and output values. For example, a function could be defined as a tool that catches specific objects (the domain) as input and makes certain other items (the range) as output. This could be a machine whereby you can get several snacks for a respective 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 cooresponds to the x-values and y-values. For instance, 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, 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 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 consider the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we cloud plug in any value for x and get itsl output value. This input set of values is required to find the range of the function f(x).

But, there are specific conditions under which a function may not be defined. For instance, if a function is not continuous at a certain point, then it is not stated 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. So, using the same function y = 2x + 1, we could see that the range would be all real numbers greater than or the same as 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 conditions under which the range may not be specified. For example, if a function is not continuous at a specific point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range could also be classified via interval notation. Interval notation explains a group of numbers using two numbers that identify the bottom and upper bounds. For instance, the set of all real numbers among 0 and 1 could be classified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be represented via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before charting 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 chart these points on a coordinate plane, it will look like this:

As we could look from the graph, the function is specified for all real numbers. This tells 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 defined for real numbers. Therefore, 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, any real number could be a possible input value. As the function only produces 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 alternates 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 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 structure y= √(ax+b) is stated just 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 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

Realize 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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