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

What are Domain and Range?

In basic terms, domain and range apply to several values in comparison to each other. For example, let's check out the grade point calculation of a school where a student gets an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the average grade. In mathematical terms, the result 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 can be stated as an instrument that takes specific objects (the domain) as input and makes certain other objects (the range) as output. This might be a tool whereby you might buy several treats for a specified amount of money.

In this piece, 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 refer to the x-values and y-values. For instance, 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, 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. In other words, 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) can be any real number because we might plug in any value for x and get a corresponding output value. This input set of values is needed to find the range of the function f(x).

Nevertheless, there are certain conditions under which a function cannot be defined. For example, 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 put it simply, it is the group of all y-coordinates or dependent variables. For instance, using the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.

Nevertheless, just as with the domain, there are particular terms under which the range may not be stated. For instance, 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 represented using interval notation. Interval notation indicates a group of numbers working with two numbers that identify the bottom and higher limits. For example, the set of all real numbers among 0 and 1 could be identified applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before creating 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 plot 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 tells 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 generates all real numbers greater than or equal to 1.

How do you figure out the Domain and Range?

The process of finding domain and range values is different for various 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 as y = ax, where a is greater than 0 and not equal to 1. For that reason, every real number can be a possible input value. As the function only 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. In addition, the function is specified 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. Consequently, the domain of the function contains all real numbers greater than or equal to b/a. A square function will consistently 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

Find 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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