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

What are Domain and Range?

In simple terms, domain and range apply to different values in in contrast to one another. For example, let's consider grade point averages 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 changes with the total score. In math, the result 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 defined as a tool that takes respective pieces (the domain) as input and makes certain other pieces (the range) as output. This can be a tool whereby you can buy multiple items for a specified amount of money.

Here, 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 cooresponds to 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 batch of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. For example, 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 might plug in any value for x and get itsl output value. This input set of values is required to figure out the range of the function f(x).

Nevertheless, there are specific cases under which a function must not be specified. 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 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 example, 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 assign to x, the output y will continue to be greater than or equal to 1.

Nevertheless, as well as with the domain, there are specific conditions under which the range may not be defined. For example, if a function is not continuous at a particular point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range could also be represented via interval notation. Interval notation explains a batch of numbers using two numbers that identify the lower and higher bounds. For instance, the set of all real numbers among 0 and 1 might be identified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented with graphs. For instance, let's review the graph of the function y = 2x + 1. Before charting a graph, we need to find 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 can look from the graph, the function is defined 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 creates 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 is different for different 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. Consequently, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. For that reason, each real number can be a possible input value. As the function only 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. 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 form y= √(ax+b) is defined just 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 consistently 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 Questions 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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