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June 10, 2022

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

## What are Domain and Range?

In basic terms, domain and range apply to multiple values in comparison to one another. For example, let's check out the grade point calculation of a school where a student gets 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 adjusts with the result. 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 instance, a function can be defined as an instrument that catches particular pieces (the domain) as input and makes particular other pieces (the range) as output. This might be a tool whereby you could get multiple items for a particular amount of money.

Today, we discuss 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 batch of all input values for the function. In other words, it is the group 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 might apply any value for x and obtain a respective output value. This input set of values is necessary to find the range of the function f(x).

Nevertheless, there are particular cases under which a function cannot be specified. For instance, if a function is not continuous at a certain 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 put it simply, it is the group of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we could see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.

However, just as with the domain, there are specific conditions under which the range may not be defined. For example, if a function is not continuous at a particular point, then it is not specified for that point.

### Domain and Range in Intervals

Domain and range could also be identified with interval notation. Interval notation expresses a batch of numbers working with two numbers that identify the bottom and upper bounds. For instance, the set of all real numbers among 0 and 1 could be represented applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be identified as follows:

(1,∞)

## Domain and Range Graphs

Domain and range can also be identified via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we have to determine 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 graph these points on a coordinate plane, it will look like this:

As we can see 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,∞).

That’s because the function generates all real numbers greater than or equal to 1.

## How do you determine the Domain and Range?

The task of finding domain and range values differs 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 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 as y = ax, where a is greater than 0 and not equal to 1. Therefore, each real number could be a possible input value. As the function just returns 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. Also, the function is defined for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Just see 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 only for x ≥ -b/a. For that reason, the domain of the function includes 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