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

What are Domain and Range?

In simple terms, domain and range refer to several values in comparison to each other. For example, let's consider 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 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 might also be thought of as input and output values. For example, a function could be defined as a machine that catches specific items (the domain) as input and generates specific other pieces (the range) as output. This could be a machine whereby you could get multiple items for a particular amount of money.

Here, we discuss 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. So, 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, 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 group of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. For instance, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we cloud plug in any value for x and obtain a respective output value. This input set of values is necessary to discover the range of the function f(x).

Nevertheless, there are certain conditions under which a function may not be stated. 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 batch 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 is all real numbers greater than or equal to 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 may not be specified. For example, if a function is not continuous at a specific point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range could also be classified using interval notation. Interval notation explains a batch of numbers applying two numbers that represent the lower and higher boundaries. For example, the set of all real numbers in the middle of 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.

Similarly, the domain and range of a function could be identified with interval notation. So, let's look at 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 specified for all real numbers.

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented using graphs. For example, let's review the graph of the function y = 2x + 1. Before charting a graph, we must 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 chart these points on a coordinate plane, it will look like this:

As we might look 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,∞).

That’s because the function generates 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. 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, every real number might be a possible input value. As the function only delivers positive values, the output of the function consists of 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 between -1 and 1. Also, the function is defined for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Just see 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. Therefore, the domain of the function consists of 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 consists of 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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