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

What are Domain and Range?

To put it simply, domain and range refer to different values in in contrast to each other. For example, let's check out the grade point calculation 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 shifts with the total score. Expressed mathematically, the score 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 might be specified as a machine that takes respective pieces (the domain) as input and makes specific other items (the range) as output. This might be a instrument whereby you could buy multiple snacks for a specified amount of money.

Here, we discuss 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 cooresponds to 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, 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. To clarify, 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 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 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 group of all possible output values for the function. In other words, 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 will 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.

But, as well as with the domain, there are specific conditions under which the range may not be specified. For example, 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 could also be classified via interval notation. Interval notation expresses a set of numbers applying two numbers that identify the bottom and higher limits. For instance, the set of all real numbers among 0 and 1 could be represented working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be represented 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 creating a graph, we have 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 graph these points on a coordinate plane, it will look like this:

As we could see 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 creates 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 is different for different types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is specified for real numbers. For that reason, 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. Consequently, each real number might be a possible input value. As the function just delivers 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 among -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]

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 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 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 Learn Functions

Grade Potential would be happy to pair you with a private math tutor if you are looking for help mastering domain and range or the trigonometric concepts. Our Alameda math tutors are practiced educators who focus on tutor you on your schedule and tailor their instruction strategy to match your learning style. Contact us today at (510) 990-9346 to learn more about how Grade Potential can support you with achieving your educational objectives.