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

What are Domain and Range?

In basic terms, domain and range coorespond with multiple values in in contrast to one another. For example, let's check out grade point averages of a school where a student receives 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 result. In mathematical terms, 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 might be stated as an instrument that catches respective pieces (the domain) as input and makes specific other pieces (the range) as output. This could be a tool whereby you could buy multiple items for a respective amount of money.

In this piece, we will teach you 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 cooresponds to the x-values and y-values. For example, 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, for 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 batch of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we might apply any value for x and get a corresponding output value. This input set of values is needed to discover the range of the function f(x).

But, there are certain conditions under which a function may not be defined. 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 batch of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equal to 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 certain conditions under which the range must not 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 could also be represented via interval notation. Interval notation indicates a batch of numbers applying two numbers that identify the bottom and upper bounds. For example, the set of all real numbers in the middle of 0 and 1 might be classified working with interval notation as follows:

(0,1)

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

Also, the domain and range of a function could be represented with 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 means 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 could also be represented using graphs. So, let's consider 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 chart these points on a coordinate plane, it will look like this:

As we could see from the graph, the function is stated 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 generates 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 differs 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 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. Consequently, each real number might be a possible input value. As the function just 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 shifts between -1 and 1. Further, the function is specified 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 specified just for x ≥ -b/a. Therefore, 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 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

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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