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

What are Domain and Range?

To put it simply, domain and range refer to multiple values in in contrast to one another. For example, let's check out the grading system of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade adjusts with the total score. In math, the score is the domain or the input, and the grade is the range or the output.

Domain and range could also be thought of as input and output values. For instance, a function might be defined as an instrument that takes specific items (the domain) as input and produces certain other pieces (the range) as output. This can be a tool whereby you might get different items for a specified quantity 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 indicate the x-values and y-values. For instance, 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 put it simply, it is the batch of all x-coordinates or independent variables. For example, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can plug in any value for x and obtain a respective output value. This input set of values is required to figure out the range of the function f(x).

However, there are particular cases under which a function cannot 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 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, using the same function y = 2x + 1, we can see that the range would be all real numbers greater than or the same as 1. Regardless of the value we assign to x, the output y will always be greater than or equal to 1.

But, just as with the domain, there are specific terms under which the range must not be specified. For example, if a function is not continuous at a certain point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range could also be identified via interval notation. Interval notation indicates a group of numbers applying two numbers that identify the lower and higher limits. For example, the set of all real numbers between 0 and 1 might be represented applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we have 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 means that the domain of the function is (-∞,∞).

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

This is because the function produces 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 various types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified for real numbers. Therefore, 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, each real number can be a possible input value. As the function just delivers 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. 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]

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 structure 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 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

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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