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

What are Domain and Range?

In simple terms, domain and range apply to multiple values in in contrast to each other. For example, let's check out 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 shifts with the total score. Expressed mathematically, 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 can be specified as an instrument that catches particular objects (the domain) as input and produces certain other items (the range) as output. This can be a tool whereby you can obtain multiple items for a specified amount of money.

In this piece, we will teach you the essentials 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. So, let's check 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 cloud apply any value for x and get a respective output value. This input set of values is necessary to discover the range of the function f(x).

But, there are specific terms under which a function must not be defined. So, if a function is not continuous at a particular point, then it is not defined 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 put it simply, it is the set of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

But, just like with the domain, there are certain terms under which the range must not be stated. For instance, if a function is not continuous at a certain point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range can also be identified using interval notation. Interval notation expresses a group of numbers working with two numbers that classify the lower and upper limits. 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 less than 1 are included in this set.

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

(-∞,∞)

This means that the function is defined for all real numbers.

The range of this function could be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be represented using graphs. For example, let's review the graph of the function y = 2x + 1. Before charting a graph, we need 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 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 tells us that the domain of the function is (-∞,∞).

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

This is because the function creates 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 various types of functions. Let's take a look at 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 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. Therefore, every real number can be a possible input value. As the function just returns 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 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]

Take a 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 specified only for x ≥ -b/a. Consequently, the domain of the function consists of 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 contains 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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