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.
🚀 Result 🚀
👉 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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