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

What are Domain and Range?

In basic terms, domain and range coorespond with different values in in contrast to each other. For instance, let's consider grade point averages of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the total score. Expressed mathematically, 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 example, a function could be specified as an instrument that takes specific pieces (the domain) as input and generates certain other objects (the range) as output. This might be a machine whereby you might buy several items for a specified quantity of money.

Today, 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. 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, 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 set of all input values for the function. To put it simply, it is the batch 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) can be any real number because we can apply any value for x and get itsl output value. This input set of values is needed to figure out the range of the function f(x).

Nevertheless, there are particular terms under which a function must not be specified. For instance, if a function is not continuous at a particular point, then it is not specified for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. For instance, working with the same function y = 2x + 1, we can see that the range would be all real numbers greater than or equal to 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range can also be classified via interval notation. Interval notation explains a batch of numbers working with two numbers that identify the lower and upper limits. For example, the set of all real numbers between 0 and 1 might be identified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

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

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

This is because the function generates all real numbers greater than or equal to 1.

How do you determine the Domain and Range?

The process of finding domain and range values differs for different 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 specified for real numbers. Consequently, the domain for an absolute value function contains 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, every 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 among -1 and 1. In addition, 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 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 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 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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