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

What are Domain and Range?

To put it simply, domain and range coorespond with different values in comparison to each other. For instance, let's consider the grade point calculation of a school where a student gets 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 result. In mathematical terms, 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 specific objects (the domain) as input and produces specific other objects (the range) as output. This can be a machine whereby you might get multiple treats for a particular amount of money.

In this piece, we discuss 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 group of all input values for the function. In other words, it is the set of all x-coordinates or independent variables. For instance, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we cloud apply any value for x and acquire a corresponding output value. This input set of values is needed to discover the range of the function f(x).

But, there are certain terms under which a function cannot be stated. For instance, 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 group 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 would be all real numbers greater than or the same as 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range could also be represented via interval notation. Interval notation indicates a group of numbers using two numbers that represent the bottom and upper boundaries. For instance, the set of all real numbers between 0 and 1 could be represented working with interval notation as follows:

(0,1)

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

Also, the domain and range of a function can be represented via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:

(-∞,∞)

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

The range of this function might be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be classified using graphs. For example, let's review the graph of the function y = 2x + 1. Before charting a graph, we must 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 chart these points on a coordinate plane, it will look like this:

As we could watch 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 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 is different for various types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is defined 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. For that reason, any real number can be a possible input value. As the function only 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 alternates among -1 and 1. Also, the function is defined 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 defined just for x ≥ -b/a. Consequently, the domain of the function contains 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 consists of 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

Find 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

Let Grade Potential Help You Excel With Functions

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