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

What are Domain and Range?

In simple terms, domain and range coorespond with multiple values in comparison to each other. For instance, let's check out the grading system of a school where a student receives an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the result. Expressed mathematically, the total 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 can be stated as a tool that catches particular pieces (the domain) as input and produces specific other objects (the range) as output. This can be a machine whereby you might get multiple snacks for a respective amount of money.

Today, we discuss the basics 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 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 batch of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can apply 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).

But, there are specific cases under which a function may not be stated. So, if a function is not continuous at a particular point, then it is not defined for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. So, using the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range might also be identified via interval notation. Interval notation explains a batch of numbers applying two numbers that classify the lower and higher bounds. For example, the set of all real numbers in the middle of 0 and 1 might be identified using interval notation as follows:

(0,1)

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

Equally, the domain and range of a function could be represented by applying interval notation. So, let's consider 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 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 represented using graphs. So, let's review the graph of the function y = 2x + 1. Before plotting a graph, we need to 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 might watch from the graph, the function is specified for all real numbers. This shows us that the domain of the function is (-∞,∞).

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

This is because the function generates 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 differs for different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is defined 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, every real number could be a possible input value. As the function just produces positive values, the output of the function contains 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. Further, the function is stated for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Just see 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 specified only for x ≥ -b/a. Therefore, 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 Examples on Domain and Range

Find 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 match you with a private math teacher if you are interested in assistance understanding domain and range or the trigonometric concepts. Our Austin math tutors are practiced professionals who focus on work with you when it’s convenient for you and tailor their tutoring techniques to fit your needs. Contact us today at (512) 387-8726 to learn more about how Grade Potential can support you with obtaining your academic goals.