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

What are Domain and Range?

To put it simply, domain and range apply to different values in in contrast to each other. For instance, 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 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 instance, a function might be defined as a machine that catches respective items (the domain) as input and produces specific other items (the range) as output. This could be a machine whereby you might buy multiple treats for a specified quantity of money.

In this piece, we review 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 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, 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 set of all input values for the function. To clarify, it is the set of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we cloud apply any value for x and get a respective output value. This input set of values is necessary to figure out the range of the function f(x).

Nevertheless, there are specific terms under which a function must not be stated. For instance, if a function is not continuous at a specific point, then it is not stated 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 set of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.

Nevertheless, just like with the domain, there are specific terms under which the range cannot be defined. For example, if a function is not continuous at a specific point, then it is not defined for that point.

Domain and Range in Intervals

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

(0,1)

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

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

(-∞,∞)

This reveals that the function is specified for all real numbers.

The range of this function could be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be represented with graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting 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 might see 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,∞).

That’s because the function produces all real numbers greater than or equal to 1.

How do you determine the Domain and Range?

The task of finding domain and range values differs 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 defined for real numbers. Consequently, the domain for an absolute value function contains 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. For that reason, each real number might be a possible input value. As the function just returns 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. Also, the function is stated 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. Consequently, 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

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