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

What are Domain and Range?

To put it simply, domain and range apply to multiple values in in contrast to each other. For example, let's check out grade point averages 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 adjusts with the total score. 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 instance, a function can be defined as a machine that takes particular pieces (the domain) as input and produces specific other objects (the range) as output. This can be a tool whereby you can obtain several items for a respective amount of money.

Here, we will teach you 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 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, whereas 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 clarify, it is the set of all x-coordinates or independent variables. So, 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 might plug in 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).

Nevertheless, there are certain cases under which a function cannot be defined. So, 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 group of all possible output values for the function. To be specific, 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 would be all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.

Nevertheless, just like with the domain, there are specific conditions 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 could also be identified with interval notation. Interval notation expresses a batch of numbers using two numbers that represent the bottom and higher boundaries. For example, the set of all real numbers between 0 and 1 might be identified working with interval notation as follows:

(0,1)

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

Similarly, the domain and range of a function can be identified using 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 means that the function is specified for all real numbers.

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

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

As we could see from the graph, the function is defined for all real numbers. This tells us that the domain of the function is (-∞,∞).

The range of the function is also (1,∞).

This is due to the fact that the function generates all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The task of finding domain and range values differs for multiple types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is defined 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, every real number can be a possible input value. As the function just produces 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. Also, the function is stated for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Take a 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 defined only for x ≥ -b/a. For that reason, the domain of the function includes all real numbers greater than or equal to b/a. A square function 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

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

Let Grade Potential Help You Excel With Functions

Grade Potential would be happy to set you up with a 1:1 math tutor if you are looking for support understanding domain and range or the trigonometric concepts. Our Marietta math tutors are experienced professionals who focus on work with you on your schedule and customize their tutoring strategy to match your needs. Reach out to us today at (770) 637-7308 to learn more about how Grade Potential can help you with achieving your learning goals.