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

What are Domain and Range?

In basic terms, domain and range coorespond with several values in in contrast to each other. For example, let's take a look at grade point averages of a school where a student receives 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 adjusts with the total score. In mathematical terms, the total 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 machine that catches specific pieces (the domain) as input and generates particular other items (the range) as output. This can be a tool whereby you might obtain several treats for a specified quantity of money.

Today, we discuss the basics 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. So, 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 batch of all input values for the function. To clarify, it is the batch of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can plug in any value for x and get itsl output value. This input set of values is needed to figure out the range of the function f(x).

Nevertheless, there are certain cases under which a function cannot be stated. For example, if a function is not continuous at a certain point, then it is not defined 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, using 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 continue to be greater than or equal to 1.

However, just like with the domain, there are certain conditions under which the range cannot be defined. For instance, if a function is not continuous at a certain point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range can also be identified via interval notation. Interval notation indicates a batch of numbers working with two numbers that identify the bottom and higher bounds. For instance, the set of all real numbers in the middle of 0 and 1 might be classified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be identified via graphs. For example, let's review the graph of the function y = 2x + 1. Before plotting 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 graph these points on a coordinate plane, it will look like this:

As we could look 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 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 different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is stated 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Therefore, every real number could be a possible input value. As the function only produces 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 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 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 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 consistently 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 Questions 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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