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

What are Domain and Range?

To put it simply, domain and range apply to several values in in contrast to one another. For instance, let's consider the grading system of a school where a student receives 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 result. In mathematical terms, the total 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 items (the domain) as input and makes specific other objects (the range) as output. This might be a machine whereby you might obtain several snacks for a particular amount 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 cooresponds to the x-values and y-values. For example, 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, 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. To put it simply, it is the batch of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and get a corresponding output value. This input set of values is needed to discover the range of the function f(x).

But, there are specific terms under which a function may not be stated. For instance, if a function is not continuous at a specific point, then it is not stated for that point.

The Range of a Function

The range of a function is the set of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we might see that the range would be all real numbers greater than or equivalent tp 1. Regardless of the value we plug in for x, the output y will continue to be greater than or equal to 1.

However, just as with the domain, there are particular conditions under which the range may not be stated. For example, 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 might also be classified using interval notation. Interval notation indicates a group of numbers working with two numbers that identify the bottom and upper boundaries. For instance, the set of all real numbers in the middle of 0 and 1 can be identified using interval notation as follows:

(0,1)

This reveals that all real numbers greater than 0 and lower than 1 are included in this batch.

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

(-∞,∞)

This reveals that the function is specified for all real numbers.

The range of this function could be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be classified with graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must 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 might watch from the graph, the function is stated for all real numbers. This means 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 multiple 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 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, any real number could be a possible input value. As the function just delivers 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. In addition, 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 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 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

Discover 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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