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 in contrast to each other. For example, let's check out 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 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 might also be thought of as input and output values. For instance, a function might be specified as a machine that takes particular items (the domain) as input and generates particular other items (the range) as output. This can be a tool whereby you could buy several snacks for a respective quantity 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 indicate the x-values and y-values. So, 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, 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 group of all input values for the function. To put it simply, it is the group 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) could be any real number because we can apply any value for x and get a corresponding output value. This input set of values is necessary to find the range of the function f(x).

But, there are specific cases under which a function cannot be defined. So, 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 batch of all possible output values for the function. To put it simply, 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 equal to 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.

However, just as with the domain, there are specific terms under which the range must not be defined. For example, if a function is not continuous at a certain point, then it is not defined 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 working with two numbers that represent the bottom and upper bounds. For instance, the set of all real numbers between 0 and 1 can 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 group.

Equally, the domain and range of a function could be identified by applying 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 reveals that the function is stated for all real numbers.

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented with 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 plot these points on a coordinate plane, it will look like this:

As we could see from the graph, the function is defined for all real numbers. This shows us that the domain of the function is (-∞,∞).

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

That’s because 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 different types of functions. Let's take a look at some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is stated 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, each 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 among -1 and 1. Further, 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 structure y= √(ax+b) is defined just 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 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 would be happy to connect you with a private math instructor if you need help comprehending domain and range or the trigonometric subjects. Our Ann Arbor math tutors are practiced professionals who focus on work with you on your schedule and customize their tutoring strategy to suit your learning style. Call us today at (734) 875-9383 to hear more about how Grade Potential can help you with obtaining your learning goals.