**What is a function in Math?**

A function is just like a machine that takes input and gives an output.

To understand this concept lets take an example of the polynomial: { x }^{ 2 }.

Now think { x }^{ 2 } is a machine.

In this machine, we put some inputs (say x) and we will see the outputs (say y).

Input (x) | Relation ({ x }^{ 2 }=x\times x) | Output (y) |
---|---|---|

7 | 7\times 7 | 49 |

2 | 2\times 2 | 4 |

0 | 0\times 0 | 0 |

-2 | (-2)\times (-2) | 4 |

-5 | (-5)\times (-5) | 25 |

Commonly we write a function as y=f(x) i.e., y as a function of x.

Now look at the results on the above table.

Here we give inputs x and the function (i.e. the machine/ relationship) squares the number inside ( ) and we get the value (i.e., output as y) of the function.

If we take x=7 then we get y=f(x)=f(7)={ x }^{ 2 }={ 7 }^{ 2 }=7\times 7=49 .

Here the function squares the number 7 and the value of the function becomes 49.

The same happens for other inputs 2, 0, -2, -5.

Table of contents – What you will learn?

## Function definition

In a simple word the answer to the question “**What is a function in Math?**” is:

A function is a rule or correspondence by which each element x is associated with a unique element y.

Let A and B be two non-empty sets of real numbers.

Let f be a rule or correspondence by which each element x\epsilon A is associates with a unique element y\epsilon B, then f is called a function defined on A into B and we write y=f(x).

## Understand the function definition

A relation f between A and B is shown on the image given below.

We will verify that this relation is a function or not by using the definition of the function.

Here

- 7 is related to 49,
- 2 is related to 4,
- 0 is related to 0,
- -2 is related to 4,
- -5 is related to 25.

You can see that element 4 of the set B is associated with two different numbers 2 and -2 of the set A.

In other words, both the numbers 2 and -2 are related to the same number 4.

Is it confusing?

Are they obeying the function definition?

Yes, both 2 and -2 are obeying the function definition as each of them corresponds to a unique number i.e. 4.

If 2 corresponds with two different numbers (say 4 and 8) then it disobeys the rule of function.

But here 2 corresponds with a unique element (not with two or more elements) and follows the rule of function.

The same happens with the element ‘-2‘.

Therefore the above relation is a function.

๐ Result ๐

๐ Two or more different elements from the set A can correspond to an element of set B.

## What is function notation?

Common **function notation** is y=f(x) and we read this as f of x.

We can write the above function as f(x)=x^{2}

Sometimes we notify the function as g(x), \: h(x), \: f(\theta ),\: g(u) etc.

For example,

- g(x)=x+1 ,
- h(x)=2x-7 ,
- f(\theta )=\sin { \theta } ,
- g(u) = 2u^{2}-7u+3 .

Later we will deeply discuss different examples of function with different factors.

## Dependent and Independent variable

In the beginning of this discussion we took the function y=f(x)=x^{2} . Here the functional value y=f(x) is dependent on x.

In mathematics x is called the **independent variable** and y is called the **dependent variable**.

Some more examples:

Function | Dependent variable | Independent variable |
---|---|---|

y=x | y | x |

z=y^{2}+1 | z | y |

v=2u^{3}+7u^{2}-4u+11 | v | u |

y=sin\frac{3\theta }{2},\frac{-\pi}{2}< \theta < \frac{\pi}{2} | y | \theta |

z=2u^{3} | z | u |

## Domain, Codomain, and Range of a function

You can see in the above image there are two sets A={-5, -2, 0, 2, 7} and B={0, 2, 4, 25, 49}.

What is the **domain, codomain, and**** range** of the function f?

Here f(-5)=25, f(-2)=4, f(0)=0, f(2)=4, f(7)=49 i,e, all the elements of set A are correspond to a unique element of set B i.e., f(A)={0, 4, 25, 49} and the element 2 of B is not related to any element of A.

Here the set A={-5, -2, 0, 2, 7} is called domain of the function f , set f(A)={0, 4, 25, 49} (a subset of set B) is called range of the function f and set B={0, 2, 4, 25, 49} is called codomain of the function f .

The set f(A)={f(x) : \forall x\epsilon \mathbb{R}} is a subset of B, and is called the range of f denoted by f(A).

With the concept of domain and codomain, we symbolically write the function f:A\rightarrow B ( f maps A into B).

In later discussion we will denote domain as the capital letter D.

## This relation is a function

Example 1

This relation is an example of function because each element of set A is related to a unique element of set B, i.e.,

- element ‘a’ of set A is related to element ‘1’ of set B,
- element ‘b’ of set A is related to element ‘2’ of set B,
- element ‘c’ of set A is related to element ‘3’ of set B,
- element ‘d’ of set A is related to element ‘4’ of set B,
- every element of the set A is related to a unique element of set B and no element of A has more than one relationship.

You can notice in the above function that each one element of set A is related to one single element of set B. This kind of function is known as **one to one function**.

Example 2

The above example of a relation is the same as of example 1 except there are two extra elements ‘5’ and ‘6’ are given in set B and no elements of set A are related to these elements.

But it is clear that all the elements of set A are related to a unique element of set B as we mentioned in example 1.

Hence the relation given in example 2 follows the rule and definition of function and consequently it is a function (no matter there are two extra elements in set B with no relationship to any element of set A).

This type of function is also called “**one to one function**“.

Example 3

Here each element of set A is related to a unique element of set B, i.e.,

- element ‘a’ of set A is related to element ‘1’ of set B,
- element ‘b’ of set A is related to element ‘1’ of set B,
- element ‘c’ of set A is related to element ‘2’ of set B,
- element ‘d’ of set A is related to element ‘3’ of set B.

You can notice that element ‘a’ and ‘b’ are related to a single element ‘1’ of set B.

But this is not a violation of the definition of function because every element of the set A is related to a unique element of set B and no element of A has more than one relationship.

Therefore the above relation is an example of a function.

The above kind of function is known as “**many to one function**“.

Here the term ‘many’ refers to the elements of set A and ‘one’ refers to the elements of set B.

Example 4

This relation is also an example of “**many to one function**“.

This is a similar example like example 2 and 3 as there are two elements ‘4’ and ‘5’ and no elements of the set A are related to these elements.

But this fact does not violets the definition of function because every element of set A is related to a unique element of set B and no element of set A has more than one relationship.

## What is not a function?

There are some relations that does not obey the rule of a function. These relations are not Function.

The examples given below are of that kind.

Example 1

Look at the above relation.

Are you thinking this is an example of one to one function?

Then observe these six points

- ‘a’ is not related to any element of B.
- ‘b’ of A is related to ‘1’ of B,
- ‘c’ of A is related to ‘2’ of B,
- ‘d’ of A is related to ‘3’ of B,
- ‘e’ of A is related to ‘4’ of B,
- ‘k’ is not related to any element of B.

The elements b, c, d, and e obeys the definition of function.

But the elements ‘a’ and ‘k’ of set A are not related to any element of set B and this fact violets the function definition.

Therefore this relation is not a function.

Example 2

Like the above relation. This is also not a function because

- the element ‘a’ of A is not related to any element of B,
- also the element ‘k’ is not related to any element of B.

Example 3

Look at the relation of the elements below:

- the element ‘a’ of A is related to two different elements ‘1’ and ‘2’ of B,
- the element ‘b’ of A is related to a unique element ‘3’ of B,
- the element ‘c’ of A is related to a unique element ‘4’ of B.

In the above-mentioned relation 2nd and 3rd points obeys the definition of a function.

But the 1st point does not obey the rule of function as we know an element of set A can not correspond to more than one element of set B and ‘a’ corresponds to two different elements of B and breaks the rule of uniqueness.

Therefore the relation is not a function.

## Ordered pair

Let us again take the function y=f(x)=x^{2}.

Here x is input (independent variable) and y is output (dependent variable).

The **ordered pair** is written as

(input, output)

where input is written first and output comes second.

So it looks like

(**x**,**y**)

or ( **x**,**f(x) **)

Now if the function takes the input 0, the output comes 0 and the ordered pair looks like

**(0,0)**

If the function takes -2 as input then 4 becomes the output and the ordered pair looks like **(-2,4)**.

**(2,4)** is an ordered pair where **2 is input** and the **output is 4**.

### Set of Ordered pairs

Above discussion we get 3 ordered pairs: (0,0), (-2,4) and (2,4).

Using these 3 ordered pairs we can define the function y=f(x)=x^{2} as the set {(0,0), (-2,4), (2,4)}.

Here the set of input values is {-2, 0, 2} which is the domain of y=f(x)=x^{2}.

The set of the output values of {0, 4} which is the range of y=f(x)=x^{2}.

## Implicit and Explicit function

The division of functions into explicit and implicit usually refers to functions specified analytically:

### Explicit function

We may express y directly in terms of x (the argument) by an analytical expression in x. We then call y an **explicit function** of x

Example:

- y=x^{2},
- y=x+\sqrt{x},
- y=x-7 etc.

### Implicit function

When a function (y) is not directly written as a function x but written as a function of x and y then it is called an **Implicit function**.

Example:

- y^{2}+3xy-x^{2}=1,
- y^{2}-4x=0,
- \frac{x^{2}}{4}+\frac{y^{2}}{9}=1

### Implicit vs Explicit functions

A relation between two variables (say x and y) which is solved for either of them, can be expressed more than one explicit functions.

For example, for the function y^{2}-4x=0 can be expressed as two functions of x (taking x as independent variable and y as dependent variable i.e., y as a function of x) as

y=+2x and y=-2x

and as a function of y (taking y as independent variable and x as dependent variable i.e., x as a function of y) as

## Algebraic operations on functions with graph

Now we will discuss different **algebraic operations on function** (sum, product, scalar multiplication, and quotient) on function.

Let D\subset \mathbb{R} and f:D\rightarrow \mathbb{R} and g:A\rightarrow \mathbb{R} be two functions on D. Then

For a better understanding of the graphs given below, the graph of each function is shown with their respective color.

### 1. Sum of two functions

The sum function f+g is defined on D by

(f+g)(x) = f(x) + g(x), x\epsilon D

Example : Let {\color{Magenta} f(x)=x^{2}}, x\epsilon \mathbb{R} and {\color{Orange} g(x)=\sqrt{x}}, x\geqslant 0.

Then the sum function is {\color{Green} (f+g)(x)=x^{2}+\sqrt{x}}, x\geq 0

### 2. Product of two functions

The product function f\times g is defined on D by

f\times g(x) = f(x)\times g(x), x\epsilon D

Example : Let {\color{Magenta} f(x)=x^{2}}, x\epsilon \mathbb{R} and {\color{Orange} g(x)=\sqrt{x}}, x\geqslant 0.

Then {\color{Green} (f\times g)(x)=x^{2}\times \sqrt{x}=x^{\frac{5}{2}}}, x\geq 0

### 3. Scalar multiplication of a function

Let k\epsilon \mathbb{R} . The function kf is defined on D by

kf(x)=kf(x), x\epsilon D

Example: Let k\epsilon \mathbb{R} and {\color{Magenta} f(x)=x^{2}},x\epsilon \mathbb{R}.

Take k=3.

Then {\color{Green}3f(x)=3x^{2}},x\epsilon \mathbb{R}

### 4. The quotient of two functions

If g(x)\neq 0,x\epsilon D, the quotient \frac{f}{g} is defined on D by

\frac{f}{g}(x)=\frac{f(x)}{g(x)}, x\epsilon D

Example: Let {\color{Magenta} f(x)=x^{2}}, x\epsilon \mathbb{R} and {\color{Orange} g(x)=\sqrt{x}}, x\geqslant 0.

Then {\color{Green} \frac{f}{g}(x)=\frac{f(x)}{g(x)}=\frac{x^{2}}{\sqrt{x}}=x^{\frac{3}{2}}},x\epsilon \mathbb{R}

## Function examples

Example 1. y=x-2.

In this function

- y is the dependent variable,
- the independent variable is x,
- 2 is a constant,
- domain = \mathbb{R}, the set of real numbers,
- codomain = \mathbb{R},
- range = \mathbb{R},
- this is an explicit function.

Example 2. z=2u.

In this function

- the dependent variable is z,
- the independent variable is u,
- domain = \mathbb{R}, the set of real numbers,
- codomain = set of even real numbers,
- range = set of all even real numbers,
- this function is an explicit function.

Example 3. w=v^{2}.

In this function

- w is the dependent variable,
- v is the independent variable,
- domain = \mathbb{R}, the set of real numbers,
- codomain = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ……} = set of all square numbers,
- range = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ……} = set of all square numbers,
- explicit function.

Example 4. y=ax^{2}+bx+c

- the dependent variable is y,
- the independent variable is x,
- explicit function

Example 5. x^{2}+y^{2}=4

- this is an example of implicit function
- we can not directly say which is independent or dependent variable,
- if we write x^{2} = 4-y^{2} \: or,\: x = \pm \sqrt{4-y^{2}} (i.e., as an implicit function), then y is the independent variable and x is the dependent variable.
- again if we write y^{2} = 4-x^{2} \: or,\: y = \pm \sqrt{4-x^{2}} , then x is independent and y is dependent.

Some more examples of implicit functions are

Example 6. x^{3}+y^{3}=3axy

Example 7. xy-2^{x}+2^{y}=0

Example 7. x^{2}+y^{2}+2gx+2fy+c=0

We hope after reading this article you understand “What is a function in math”.

If you have any questions or suggestions regarding this topic feel free to ask in the comment section.

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