June 10, 2022

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

What are Domain and Range?

To put it simply, domain and range coorespond with multiple values in in contrast to one another. For instance, let's take a look at the grade point calculation 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 adjusts with the total score. In math, 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 could be specified as an instrument that takes respective objects (the domain) as input and makes specific other pieces (the range) as output. This could be a instrument whereby you could obtain different treats for a specified amount of money.

Here, we review 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 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, 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 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 review 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 obtain a respective output value. This input set of values is required to figure out the range of the function f(x).

Nevertheless, there are specific conditions under which a function may not be stated. For example, if a function is not continuous at a certain point, then it is not stated for that point.

The Range of a Function

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

However, as well as with the domain, there are certain terms under which the range cannot be defined. 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 represented with interval notation. Interval notation indicates a batch of numbers working with two numbers that identify the lower and higher limits. For example, the set of all real numbers between 0 and 1 could be classified using interval notation as follows:

(0,1)

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

Also, the domain and range of a function might be identified by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be represented 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 might also be identified via graphs. So, let's consider 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 chart these points on a coordinate plane, it will look like this:

As we can watch from the graph, the function is defined for all real numbers. This means that the domain of the function is (-∞,∞).

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

This is because the function produces 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 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. Therefore, 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. For that reason, every real number might be a possible input value. As the function just produces 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 between -1 and 1. In addition, 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 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. 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 includes 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 Excel With Functions

Grade Potential can pair you with a private math teacher if you are looking for assistance mastering domain and range or the trigonometric subjects. Our Akron math tutors are skilled educators who strive to tutor you on your schedule and customize their instruction strategy to match your learning style. Contact us today at (330) 632-3388 to learn more about how Grade Potential can support you with obtaining your academic goals.