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

What are Domain and Range?

To put it simply, domain and range coorespond with different values in comparison to each other. For example, let's consider the grading system of a school where a student gets 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 adjusts with the average grade. Expressed mathematically, the result 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 catches specific pieces (the domain) as input and produces certain other objects (the range) as output. This can be a instrument whereby you can get multiple treats for a specified amount of money.

Here, we will teach you 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 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, 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 batch of all input values for the function. To put it simply, 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) might be any real number because we cloud plug in any value for x and get itsl output value. This input set of values is needed to discover the range of the function f(x).

Nevertheless, there are certain cases under which a function must not be specified. For example, if a function is not continuous at a particular point, then it is not specified for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we could see that the range would be all real numbers greater than or the same as 1. Regardless of the value we assign to x, the output y will always be greater than or equal to 1.

However, just as with the domain, there are particular terms under which the range must not be specified. For example, if a function is not continuous at a particular point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range might also be classified using interval notation. Interval notation expresses a set of numbers using two numbers that classify the lower and upper limits. For instance, the set of all real numbers between 0 and 1 can be represented using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be classified via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before creating 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 can 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 due to the fact that the function generates all real numbers greater than or equal to 1.

How do you figure out 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 structure y=|ax+b| is defined 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. Consequently, any real number can be a possible input value. As the function only 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 varies between -1 and 1. In addition, the function is defined 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 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 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 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 Master Functions

Grade Potential would be happy to connect you with a one on one math tutor if you are interested in support mastering domain and range or the trigonometric subjects. Our Centennial math tutors are experienced professionals who focus on partner with you on your schedule and personalize their tutoring techniques to suit your learning style. Call us today at (303) 578-1372 to hear more about how Grade Potential can assist you with reaching your academic goals.