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# Domain and Range of a Function – Explanation & Examples

This article** will explain the domain and range of a function mean and how to calculate the two quantities.** Before getting into the topic of domain and range, let’s briefly describe what a function is.

**In mathematics, we can compare a function to a machine that generates some output in correlation to a given input**. By taking an example of a coin stamping machine, we can illustrate a function’s meaning as follows.

When you insert a coin into the coin stamping machine, the result is a stamped and flattened piece of metal. By considering a function, we can relate the coin and the flattened piece of metal with the domain and range. In this case, a function is considered to be the coin stamping machine.

Just like the coin stamping machine, which can only produce a single flattened piece of metal at a time, a function works in the same manner by giving out one result at a time.

## History of a function

The idea of a function was introduced in the early seventeenth century when **Rene Descartes (1596-1650)** used the concept in his book Geometry (1637) to model mathematical problems.

Fifty years later, after the publication of Geometry, Gottfried Wilhelm Leibniz (1646-1716) introduced the term “function.” Later, Leonhard Euler (1707-1783) played a big role by introducing the technique of function notion, y = f (x).

## Real-life application of a function

Functions are very useful in mathematics because they allow us to model real-life problems into a mathematical format.

*Here are a few examples of the application of a function.*

#### Circumference of a circle

The circumference of a circle is a function of its diameter or radius. We can mathematically represent this statement as:

C(d) =dπ or C(r)=2π⋅r

#### A shadow

The length of the shadow of an object is a function of its height.

#### The position of a moving object

The location of a moving object such as a car is a function of time.

#### Temperature

The temperature of a body is based on several factors and inputs.

#### Money

The compound or simple interest is a function of the time, principal, and interest rate.

#### Height of an object

The height of an object is a function of his/her age and body weight.

Having learned about a function now can proceed to how to calculate the domain and the range of a function.

## What is the Domain and Range of a Function?

The **domain of a function** is the input numbers that, when plugged into a function, the result is defined. In simple words, we can define the domain of a function as the possible values of x that will make an equation true.

Some of the instances that will not make a valid function are when an equation is being divided by zero or a negative square root.

For example, f(*x*) = *x*^{2} is a valid function because, no matter what value of x can be substituted into an equation, there is always a valid answer. For this reason, we can conclude that the domain of any function is all real numbers.

The **range of a function** is defined as a set of solutions to the equation for a given input. In other words, the range is the output or y value of a function. There is only one range for a given function.

### How to use interval notations to specify Domain and Range?

Since the range and domain of a function are usually expressed in interval notation, it’s important to discuss the concept of interval notation.

*The procedure for doing interval notation include:*

- Write the numbers separated by a comma in ascending order.
- Enclose the numbers using parentheses () to show that an endpoint value is not included.
- Use brackets [] to enclose the numbers when the endpoint value is included.

## How to Find the Domain and Range of a Function?

We can determine the domain of a function either algebraically or by the graphical method. To calculate the domain of a function algebraically, you solve the equation to determine the values of x.

Different types of functions have their own methods of determining their domain.

*Let’s examine these types of functions and how to calculate their domain.*

### How to find the domain for a function with no denominator or radicals?

Let’s see a few examples below to understand this scenario.

*Example 1*

Find the domain of f (x) = 5x − 3

__Solution__

The domain of a linear function is all real numbers, therefore,

Domain: (−∞, ∞)

Range: (−∞, ∞)

**A function with a radical **

*Example 2*

Find the domain of the function f(x)=−2x^{2 }+ 12x + 5

__Solution__

The function f(x) = −2x^{2 }+ 12x + 5 is a quadratic polynomial, therefore, the domain is (−∞, ∞)

### How to find the domain for a rational function with a variable in the denominator?

To find the domain of this type of function, set the denominator to zero and calculate the variable’s value.

Let’s see a few examples below to understand this scenario.

*Example 3*

Determine the domain of x−4/ (x^{2 }−2x−15)

__Solution__

Set the denominator to zero and solve for x

⟹ x^{2 }− 2x – 15 = (x − 5) (x + 3) = 0

Hence, x = −3, x = 5

For the denominator not to be zero, we need to avoid the numbers −3 and 5. Therefore, the domain is all real numbers except −3 and 5.

*Example 4*

Calculate the domain and the range of the function f(x) = -2/x.

__Solution__

Set the denominator to zero.

⟹ x = 0

Therefore, domain: All real numbers except 0.

The range is all real values of x except 0.

*Example 5*

Find the domain and range of the following function.

f(x) = 2/ (x + 1)

__Solution__

Set the denominator equal to zero and solve for x.

x + 1 = 0

= -1

Since the function is undefined when x = -1, the domain is all real numbers except -1. Similarly, the range is all real numbers except 0

### How to the domain for a function with a variable inside a radical sign?

To find the domain of the function, the terms inside the radical are set the inequality of > 0 or ≥ 0. Then, the value of the variable is determined.

Let’s see a few examples below to understand this scenario.

*Example 6*

Find the domain of f(x) = √ (6 + x – x^{2})

__Solution__

To avoid the square roots of negative numbers, we set the expression inside the radical sign to ≥ 0.

6 + x – x^{2} ≥ 0 ⟹ x ^{2 }– x – 6≤ 0

⟹ x ^{2 }– x – 6= (x – 3) (x +2) = 0

Therefore, the function is zero if x = 3 or x = -2

Hence the domain: [−2, 3]

*Example 7*

Find the domain of f(x) =x/√ (x^{2 }– 9)

__Solution__

Set the expression within the radical sign to x^{2 }– 9 > 0

Solve for the variable to get;

x = 3 or – 3

Therefore, Domain: (−∞, −3) & (3, ∞)

*Example 8*

Find the domain of f(x) = 1/√ (x^{2} -4)

__Solution__

By factoring the denominator, we get x ≠ (2, – 2).

Test your answer by plugging -3 into the expression within the radical sign.

⟹ (-3)^{2} – 4 = 5

also try with zero

⟹ 0^{2} – 4 = -4, therefore number between 2 and -2 are invalid

Try number above 2

⟹ 3^{2} – 4 = 5. This one is valid.

Hence, the domain = (-∞, -2) U (2, ∞)

### How to find the domain of a function using the natural logarithm (ln)?

To find the domain of a function using natural log, set the terms within the parentheses to >0 and then solve.

Let’s see an example below to understand this scenario.

*Example 9*

Find the domain of the function f(x) = ln (x – 8)

__Solution__

⟹ x – 8 > 0

⟹ x – 8 + 8 > 0 + 8

⟹ x > 8

Domain:(8, ∞)

### How to find the domain and range of a relation?

A relation is an asset of x and y coordinates. To find the domain and range in a relation, just list the x and y values, respectively.

Let’s see a few examples below to understand this scenario.

*Example 10*

State the domain and range of the relation {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

__Solution__

List the x values. Domain: {2, 3, 4, 6}

List the y values. range: {–3, –1, 3, 6}

*Example 11*

Find the domain and range of the relation {(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

__Solution__

The domain is {–3, –2, –1, 0, 1, 2} and the range is {5}

*Example 12*

Given that R = {(4, 2) (4, -2), (9, 3) (9, -3)}, find the domain and range of R.

__Solution__

The domain is a list of first values, therefore, D= {4, 9} and the range = {2, -2, 3, -3}