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

What are Domain and Range?

In simple terms, domain and range apply to multiple values in in contrast to one another. For example, let's consider 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 shifts with the average grade. In math, the total 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 could be stated as an instrument that catches particular objects (the domain) as input and makes particular other objects (the range) as output. This could be a machine whereby you might buy different items for a respective quantity 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 view 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. In other words, it is the group of all x-coordinates or independent variables. For instance, let's consider 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 obtain a corresponding output value. This input set of values is needed to discover the range of the function f(x).

Nevertheless, there are specific terms under which a function cannot be stated. For example, if a function is not continuous at a specific 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. To be specific, it is the set of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we might see that the range would be 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 specific conditions under which the range must not be defined. For example, 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 expresses a set of numbers working with two numbers that identify the lower and higher boundaries. For example, the set of all real numbers in the middle of 0 and 1 can be classified applying interval notation as follows:

(0,1)

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

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 identified as follows:

(-∞,∞)

This means 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 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 graph these points on a coordinate plane, it will look like this:

As we might see from the graph, the function is stated 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 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 watch 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 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, every real number can 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. In addition, 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 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 stated only for x ≥ -b/a. Therefore, the domain of the function contains 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 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

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 Learn Functions

Grade Potential would be happy to connect you with a 1:1 math tutor if you are looking for help comprehending domain and range or the trigonometric topics. Our Washington DC math tutors are experienced professionals who strive to partner with you when it’s convenient for you and personalize their teaching strategy to match your learning style. Contact us today at (202) 978-6204 to hear more about how Grade Potential can assist you with reaching your learning goals.