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

What are Domain and Range?

In basic terms, domain and range apply to multiple values in comparison to each other. For instance, 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 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 could also be thought of as input and output values. For example, a function can be defined as a tool that catches specific objects (the domain) as input and makes specific other items (the range) as output. This could be a machine whereby you might get multiple snacks for a particular amount of money.

In this piece, we will teach you the essentials 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 example, 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, for 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. In other words, 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) can be any real number because we cloud plug in any value for x and get a respective output value. This input set of values is required to discover the range of the function f(x).

However, there are particular cases under which a function must not be stated. 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 set of all possible output values for the function. To be specific, it is the batch of all y-coordinates or dependent variables. For instance, working with the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.

But, as well as with the domain, there are specific terms under which the range cannot be defined. 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 with interval notation. Interval notation explains a batch of numbers working with two numbers that identify the bottom and upper bounds. For example, the set of all real numbers between 0 and 1 could be classified applying interval notation as follows:

(0,1)

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

Also, the domain and range of a function can be represented with interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) might be classified as follows:

(-∞,∞)

This means that the function is specified for all real numbers.

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented using graphs. For instance, 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 can 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 due to the fact that the function produces 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 various types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified 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. Therefore, each real number can be a possible input value. As the function just returns 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 among -1 and 1. Also, 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 structure y= √(ax+b) is defined just for x ≥ -b/a. Therefore, 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 contains 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

Let Grade Potential Help You Learn Functions

Grade Potential would be happy to set you up with a private math instructor if you are looking for help understanding domain and range or the trigonometric concepts. Our Hillsboro math tutors are experienced educators who strive to partner with you on your schedule and personalize their instruction strategy to suit your learning style. Contact us today at (503) 506-6314 to hear more about how Grade Potential can assist you with reaching your educational objectives.