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

What are Domain and Range?

In simple terms, domain and range coorespond with different values in comparison to each other. For example, let's take a look at 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 total score. Expressed mathematically, the total 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 could be specified as an instrument that catches specific pieces (the domain) as input and makes particular other objects (the range) as output. This could be a instrument whereby you could buy multiple treats for a respective amount of money.

Here, we review 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 indicate the x-values and y-values. For example, 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 group 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 review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud plug in any value for x and get a corresponding output value. This input set of values is necessary to figure out the range of the function f(x).

However, there are certain terms under which a function cannot 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. To be specific, it is the set of all y-coordinates or dependent variables. For instance, 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. Regardless of the 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 conditions under which the range must not 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 with interval notation. Interval notation expresses a group of numbers applying two numbers that classify the bottom and upper bounds. For instance, the set of all real numbers in the middle of 0 and 1 could be identified applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

This reveals that the function is specified for all real numbers.

The range of this function can be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be classified with graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we must 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 look from the graph, the function is stated for all real numbers. This shows us 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 task of finding domain and range values differs for multiple types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the form 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, any real number could be a possible input value. As the function only produces 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 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 specified only for x ≥ -b/a. For that reason, the domain of the function contains 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 includes 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

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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