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 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 a cumulative score of 81 - 90, and so on. Here, the grade changes with the average grade. In mathematical terms, the result 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 defined as a machine that takes respective pieces (the domain) as input and produces particular other items (the range) as output. This might be a instrument whereby you can obtain multiple snacks for a particular quantity of money.

In this piece, we will teach you the basics 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 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, 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 set of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud apply any value for x and get a respective output value. This input set of values is needed to figure out the range of the function f(x).

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

The Range of a Function

The range of a function is the set of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. So, using the same function y = 2x + 1, we can see that the range will 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.

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

Domain and Range in Intervals

Domain and range might also be classified via interval notation. Interval notation expresses a batch of numbers using two numbers that represent the lower and higher boundaries. For instance, the set of all real numbers among 0 and 1 could be classified using interval notation as follows:

(0,1)

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

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

(-∞,∞)

This reveals that the function is defined 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 classified via graphs. So, let's consider 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 might look 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 because the function creates all real numbers greater than or equal to 1.

How do you figure out the Domain and Range?

The process of finding domain and range values is different for various 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. 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. Therefore, every real number could be a possible input value. As the function only produces 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 between -1 and 1. Further, 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 structure y= √(ax+b) is specified just for x ≥ -b/a. Therefore, the domain of the function contains 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 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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