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

What are Domain and Range?

In basic terms, domain and range coorespond with several values in in contrast to each other. For instance, let's consider the grade point calculation of a school where a student earns 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 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 a tool that catches specific objects (the domain) as input and produces certain other objects (the range) as output. This might be a machine whereby you could get different snacks for a particular quantity of money.

In this piece, we discuss the fundamentals of the domain and the range of mathematical functions.

What are 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. In other words, 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 acquire a corresponding output value. This input set of values is necessary to find the range of the function f(x).

However, there are certain cases under which a function cannot be specified. So, if a function is not continuous at a particular point, then it is not stated 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. So, applying the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.

Nevertheless, just like with the domain, there are certain terms under which the range may not be specified. For example, 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 might also be classified using interval notation. Interval notation explains a group of numbers using two numbers that identify the lower and upper boundaries. For instance, the set of all real numbers in the middle of 0 and 1 might be classified applying interval notation as follows:

(0,1)

This denotes 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 identified via interval notation. So, let's look at 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 identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented using graphs. So, let's consider the graph of the function y = 2x + 1. Before plotting 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 plot these points on a coordinate plane, it will look like this:

As we could see from the graph, the function is specified 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 creates 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 structure y=|ax+b| is stated for real numbers. For that reason, 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 might be a possible input value. As the function just produces positive values, the output of the function consists of 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]

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 structure y= √(ax+b) is specified only for x ≥ -b/a. For that reason, 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 contains 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

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