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

What are Domain and Range?

In simple terms, domain and range apply to several values in in contrast to one another. For instance, let's consider the grading system of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the total score. In mathematical terms, 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 might be specified as a tool that catches respective pieces (the domain) as input and makes specific other objects (the range) as output. This might be a machine whereby you might get different items for a specified quantity 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 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, 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 group of all input values for the function. In other words, it is the batch of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we might plug in any value for x and acquire itsl 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 specified. For instance, if a function is not continuous at a particular point, then it is not specified 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 be specific, 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 would be all real numbers greater than or the same as 1. No matter what value we apply to x, the output y will always be greater than or equal to 1.

Nevertheless, just as with the domain, there are specific terms under which the range may not be stated. 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 might also be identified via interval notation. Interval notation explains a group of numbers using two numbers that represent the lower and higher boundaries. For example, the set of all real numbers between 0 and 1 could be identified applying interval notation as follows:

(0,1)

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

Equally, the domain and range of a function could be identified with interval notation. So, let's look at 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 can also be represented via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need to determine 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 look 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 due to the fact that the function creates 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 is different for different types of functions. Let's take a look at 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. Consequently, every real number might be a possible input value. As the function only 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 alternates 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 form y= √(ax+b) is defined just for x ≥ -b/a. For that reason, the domain of the function contains 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

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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