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 comparison to each other. For instance, let's take a look at the grading system 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 shifts with the result. Expressed mathematically, 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 example, a function could be defined as a machine that takes respective items (the domain) as input and produces specific other pieces (the range) as output. This could be a instrument whereby you might obtain multiple snacks for a respective quantity of money.

In this piece, we will teach you the fundamentals 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 refer to the x-values and y-values. For instance, let's check 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 batch of all input values for the function. To put it simply, it is the set 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) can be any real number because we cloud plug in any value for x and obtain a corresponding output value. This input set of values is necessary to figure out the range of the function f(x).

Nevertheless, there are specific cases under which a function may not be specified. 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. To be specific, it is the batch of all y-coordinates or dependent variables. For instance, using the same function y = 2x + 1, we might see that the range is all real numbers greater than or the same as 1. No matter what value we apply to x, the output y will continue to be greater than or equal to 1.

Nevertheless, as well as with the domain, there are certain conditions under which the range cannot be stated. For instance, if a function is not continuous at a particular point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range can also be identified using interval notation. Interval notation expresses a batch of numbers working with two numbers that represent the lower and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 could be classified applying interval notation as follows:

(0,1)

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

Equally, the domain and range of a function might be identified using interval notation. So, let's consider 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 might be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented with graphs. So, let's review the graph of the function y = 2x + 1. Before plotting 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 watch from the graph, the function is stated for all real numbers. This shows 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 determine the Domain and Range?

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

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified for real numbers. For that reason, 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, each real number can be a possible input value. As the function just returns 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 oscillates 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]

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 form y= √(ax+b) is specified 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

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 Master Functions

Grade Potential would be happy to pair you with a one on one math teacher if you are interested in help mastering domain and range or the trigonometric subjects. Our Mountain View math tutors are skilled educators who focus on work with you when it’s convenient for you and personalize their teaching techniques to fit your needs. Reach out to us today at (650) 459-5193 to hear more about how Grade Potential can assist you with reaching your learning goals.