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

What are Domain and Range?

In basic terms, domain and range apply to multiple values in comparison to one another. For instance, let's take a look at grade point averages of a school where a student earns 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 changes with the average grade. In math, the score 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 instance, a function could be defined as a tool that catches specific items (the domain) as input and makes particular other objects (the range) as output. This might be a machine whereby you might get several items for a specified quantity of money.

Here, we discuss the basics of the domain and the range of mathematical functions.

What is the Domain and Range of a Function?

In algebra, the domain and the range refer to the x-values and y-values. So, 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, 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 batch of all input values for the function. To put it simply, it is the batch of all x-coordinates or independent variables. For instance, let's consider the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and obtain itsl output value. This input set of values is required to discover the range of the function f(x).

However, there are certain cases under which a function must not 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. To put it simply, it is the batch of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range would be all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

However, as well as with the domain, there are certain terms under which the range must not be defined. For instance, 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 can also be classified using interval notation. Interval notation expresses a batch of numbers working with two numbers that classify the bottom and higher limits. For example, the set of all real numbers in the middle of 0 and 1 might be identified using interval notation as follows:

(0,1)

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

Equally, the domain and range of a function might be represented by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:

(-∞,∞)

This tells us that the function is defined for all real numbers.

The range of this function might be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified via graphs. So, let's review the graph of the function y = 2x + 1. Before plotting 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 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 find the Domain and Range?

The process of finding domain and range values is different for different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is defined for real numbers. Therefore, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Therefore, every real number can be a possible input value. As the function just delivers 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 oscillates among -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 structure y= √(ax+b) is specified just 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 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

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