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 comparison to each other. For instance, let's take a look at grade point averages 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 changes with the total score. Expressed mathematically, 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 example, a function could be defined as a machine that catches particular objects (the domain) as input and generates particular other pieces (the range) as output. This can be a instrument whereby you could buy multiple treats for a respective quantity of money.

Here, we will teach you 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 example, 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, for 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 set of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. So, 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 can apply any value for x and get itsl output value. This input set of values is necessary to discover the range of the function f(x).

However, there are particular conditions under which a function may not be specified. For instance, if a function is not continuous at a specific 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 be specific, it is the batch of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we could see that the range is all real numbers greater than or equal to 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

However, just like with the domain, there are certain conditions under which the range cannot be defined. For example, 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 could also be classified using interval notation. Interval notation expresses a batch of numbers using two numbers that classify the bottom and upper limits. For instance, the set of all real numbers in the middle of 0 and 1 might be identified using interval notation as follows:

(0,1)

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

Also, the domain and range of a function could be classified using 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 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 can also be represented using graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we have to 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 plot these points on a coordinate plane, it will look like this:

As we might see 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 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 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 stated for real numbers. Therefore, 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. Consequently, every real number can be a possible input value. As the function just 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 alternates between -1 and 1. Further, 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 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 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 consistently 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 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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