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 in contrast to one another. 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 shifts 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 instance, a function might be defined as a tool that takes specific objects (the domain) as input and makes particular other pieces (the range) as output. This can be a instrument whereby you might buy different treats for a particular quantity of money.

Today, 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 look at 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, whereas 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 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 acquire itsl output value. This input set of values is necessary to discover the range of the function f(x).

However, there are specific terms under which a function must not be specified. So, if a function is not continuous at a certain point, then it is not defined 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. So, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

Nevertheless, just as with the domain, there are certain terms under which the range cannot be specified. For instance, 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 explains a group of numbers using two numbers that classify the lower and upper boundaries. For instance, the set of all real numbers among 0 and 1 could be represented working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified using graphs. For example, let's consider 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 might 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 because the function produces all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The task 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 stated 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. Therefore, each real number can be a possible input value. As the function just 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 alternates between -1 and 1. Also, 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 structure y= √(ax+b) is defined 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 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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