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

What are Domain and Range?

In basic terms, domain and range refer 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 a cumulative score of 81 - 90, and so on. Here, the grade changes with the average grade. 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 example, a function might be defined as a tool that catches specific objects (the domain) as input and generates particular other objects (the range) as output. This might be a instrument whereby you might obtain different treats for a respective quantity of money.

In this piece, we will teach you the basics 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. For instance, 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 group of all input values for the function. To clarify, it is the set 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) might be any real number because we cloud apply any value for x and obtain itsl output value. This input set of values is needed to figure out the range of the function f(x).

But, there are certain cases under which a function must not be defined. So, 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 set of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. So, working with 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 assign to x, the output y will always be greater than or equal to 1.

But, just as with the domain, there are specific conditions under which the range must not be specified. 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 could also be identified with interval notation. Interval notation indicates a group of numbers using two numbers that represent the bottom and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 might be represented 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 could be represented by applying interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:

(-∞,∞)

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

The range of this function could be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be identified via graphs. So, let's consider the graph of the function y = 2x + 1. Before charting 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 chart these points on a coordinate plane, it will look like this:

As we might see from the graph, the function is specified for all real numbers. This tells us 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 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 structure y=|ax+b| is specified 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, each real number might be a possible input value. As the function just produces 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 shifts among -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 stated just for x ≥ -b/a. Consequently, the domain of the function contains all real numbers greater than or equal to b/a. A square function will 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

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

Let Grade Potential Help You Learn Functions

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