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

What are Domain and Range?

To put it simply, domain and range apply to multiple values in in contrast to each other. For instance, let's consider the grading system of a school where a student receives 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 average grade. In math, the total is the domain or the input, and the grade is the range or the output.

Domain and range might also be thought of as input and output values. For example, a function might be specified as a machine that takes specific objects (the domain) as input and produces certain other items (the range) as output. This might be a tool whereby you can get multiple snacks for a particular quantity of money.

Today, 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 refer 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 batch of all input values for the function. To put it simply, it is the set of all x-coordinates or independent variables. For instance, 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 plug in any value for x and obtain a respective output value. This input set of values is necessary to figure out the range of the function f(x).

However, there are certain cases under which a function may not be specified. For instance, if a function is not continuous at a specific point, then it is not defined 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 batch of all y-coordinates or dependent variables. For instance, using the same function y = 2x + 1, we could see that the range will be all real numbers greater than or equivalent tp 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

Nevertheless, as well as with the domain, there are certain conditions under which the range may not be defined. For instance, 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 could also be classified with interval notation. Interval notation indicates a set of numbers working with two numbers that represent the lower and upper limits. For instance, the set of all real numbers between 0 and 1 can be classified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function could be classified as follows:

(1,∞)

Domain and Range Graphs

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

As we could see 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 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 multiple types of functions. Let's consider 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 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 as 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 among -1 and 1. Also, 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 structure y= √(ax+b) is stated only for x ≥ -b/a. Consequently, the domain of the function includes 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

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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