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 comparison to one another. For instance, let's consider the grade point calculation of a school where a student receives 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 adjusts 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 example, a function can be defined as an instrument that catches respective objects (the domain) as input and makes certain other items (the range) as output. This could be a instrument whereby you might get different treats for a specified amount of money.

Here, we review the fundamentals 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 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, 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 set of all input values for the function. To put it simply, it is the set 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) could be any real number because we can apply any value for x and obtain itsl output value. This input set of values is needed to find the range of the function f(x).

Nevertheless, there are particular conditions under which a function must not be defined. So, if a function is not continuous at a certain point, then it is not stated for that point.

The Range of a Function

The range of a function is the set of all possible output values for the function. In other words, it is the batch of all y-coordinates or dependent variables. For example, working with 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.

Nevertheless, just like with the domain, there are particular terms under which the range cannot be defined. For example, if a function is not continuous at a particular point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range can also be identified via interval notation. Interval notation explains a set of numbers applying two numbers that identify the bottom and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 can be represented applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function could be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be identified via graphs. For instance, 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 graph 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 tells us that the domain of the function is (-∞,∞).

The range of the function is also (1,∞).

That’s because 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 differs for different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Consequently, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Therefore, any real number might 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 varies 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 stated just for x ≥ -b/a. For that reason, 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 Examples 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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