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June 10, 2022

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

## What are Domain and Range?

In simple terms, domain and range coorespond with different values in in contrast to each other. For example, let's check out the grade point calculation of a school where a student receives 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 changes with the result. Expressed mathematically, the score 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 could be defined as an instrument that takes specific pieces (the domain) as input and generates specific other pieces (the range) as output. This can be a machine whereby you might buy several treats for a particular quantity of money.

In this piece, we review the essentials 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 cooresponds 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, 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 instance, let's consider 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 get itsl output value. This input set of values is needed to figure out the range of the function f(x).

However, there are certain conditions under which a function must not be specified. For instance, if a function is not continuous at a particular 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 set of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.

But, as well as with the domain, there are certain conditions under which the range may not be defined. 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 might also be represented via interval notation. Interval notation expresses a group of numbers working with two numbers that classify the bottom and upper bounds. For instance, the set of all real numbers between 0 and 1 can be represented working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function can be classified as follows:

(1,∞)

## Domain and Range Graphs

Domain and range might also be identified using graphs. For instance, 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 watch 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,∞).

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

## How do you find the Domain and Range?

The process of finding domain and range values differs for different types of functions. Let's watch 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 consists of 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, any real number can be a possible input value. As the function only delivers 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 varies 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 specified only for x ≥ -b/a. Therefore, the domain of the function consists of 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 includes 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

Discover 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   