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

What are Domain and Range?

In basic terms, domain and range refer to multiple values in in contrast to each other. For instance, let's take a look at the grading system of a school where a student earns an A grade for an average between 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 total 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 example, a function can be specified as a machine that takes respective objects (the domain) as input and produces particular other pieces (the range) as output. This can be a machine whereby you might get different snacks for a specified quantity of money.

In this piece, we will teach you 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 cooresponds 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, 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. In other words, it is the batch of all x-coordinates or independent variables. For instance, let's review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can plug in any value for x and get itsl output value. This input set of values is necessary to discover the range of the function f(x).

But, there are certain conditions under which a function may not be specified. So, if a function is not continuous at a certain 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 example, using the same function y = 2x + 1, we can see that the range is 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.

However, as well as with the domain, there are particular conditions under which the range cannot be stated. For example, if a function is not continuous at a specific point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range might also be classified via interval notation. Interval notation indicates a batch of numbers working with two numbers that identify the lower and upper boundaries. For instance, the set of all real numbers among 0 and 1 can 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 set.

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

(-∞,∞)

This tells us 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 identified with graphs. So, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we need to find 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 graph these points on a coordinate plane, it will look like this:

As we could watch 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 produces all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The task of finding domain and range values differs for multiple 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. Therefore, 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, any real number could be a possible input value. As the function only produces 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 between -1 and 1. In addition, the function is defined 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 defined just for x ≥ -b/a. Consequently, the domain of the function includes 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 Questions on Domain and Range

Realize 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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