# Functions in Discrete Mathematics

Functions are an important part of discrete mathematics. This article is all about functions, their types, and other details of functions. A function assigns exactly one element of a set to each element of the other set. Functions are the rules that assign one input to one output. The function can be represented as f: A **⇢** B. A is called the* domain of the function* and B is called the* codomain function*.

### Functions:

- A function assigns exactly one element of one set to each element of other sets.
- A function is a rule that assigns each input exactly one output.
- A function f from A to B is an assignment of exactly one element of B to each element of A (where A and B are non-empty sets).
- A function f from set A to set B is represented as
**f: A ⇢ B**where A is called the domain of f and B is called as codomain of f. - If b is a unique element of B to element a of A assigned by function F then, it is written as f(a) = b.
- Function f maps A to B means f is a function from A to B i.e. f: A
**⇢**B

### Domain of a function:

- If f is a function from set A to set B then, A is called the domain of function f.
- The set of all inputs for a function is called its domain.

### Codomain of a function:

- If f is a function from set A to set B then, B is called the codomain of function f.
- The set of all allowable outputs for a function is called its codomain.

### Pre-image and Image of a function:

A function f: A **⇢** B such that for each a ∈ A, there exists a unique b ∈ B such that (a, b) ∈ R then, a is called the pre-image of f and b is called the image of f.

### Types of function:

### One-One function ( or Injective Function):

A function in which one element of the domain is connected to one element of the codomain.

A function f: A **⇢** B is said to be a one-one (injective) function if different elements of A have different images in B.

**f: A ⇢ B is one-one **

**⇒ a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A**

**⇒ f(a) = f(b) ⇒ a = b for all a, b ∈ A**

### Many-One function:

A function f: A **⇢** B is said to be a many-one function if two or more elements of set A have the same image in B.

A function f: A **⇢** B is a many-one function if it is not a one-one function.

**f: A ⇢ B is many-one **

**⇒ a ≠ b but f(a) = f(b) for all a, b ∈ A**

### Onto function( or Surjective Function):

A function f: A -> B is said to be onto (surjective) function if every element of B is an image of some element of A i.e. f(A) = B or range of f is the codomain of f.

A function in which every element of the codomain has one pre-image.

**f: A ⇢ B is onto if for each b∈ B, there exists a∈ A such that f(a) = b.**

### Into Function:

A function f: A **⇢** B is said to be an into a function if there exists an element in B with no pre-image in A.

A function f: A**⇢** B is into function when it is not onto.

### One-One Correspondent function( or Bijective Function or One-One Onto Function):

A function which is both one-one and onto (both injective and surjective) is called one-one correspondent(bijective) function.

**f : A ⇢ B is one-one correspondent (bijective) if:**

**one-one i.e. f(a) = f(b) ⇒ a = b for all a, b ∈ A****onto i.e. for each b ∈ B, there exists a ∈ A such that f(a) = b.**

### One-One Into function:

A function that is both one-one and into is called one-one into function.

### Many-one onto function:

A function that is both many-one and onto is called many-one onto function.

### Many-one into a function:

A function that is both many-one and into is called many-one into function.

### Inverse of a function:

Let f: A **⇢** B be a bijection then, a function g: B **⇢ **A which associates each element b ∈ B to a different element a ∈ A such that f(a) = b is called the inverse of f.

**f(a) = b ↔︎ g(b) = a**

### Composition of functions :-

Let f: A **⇢** B and g: B **⇢ **C be two functions then, a function gof: A **⇢ **C is defined by

**(gof)(x) = g(f(x)), for all x ∈ A **

is called the composition of f and g.

### Note:

Let X and Y be two sets with m and n elements and a function is defined as f : X->Y then,

- Total number of functions = n
^{m} - Total number of one-one function =
^{ n}P_{m} - Total number of onto functions = n
^{m}–^{n}C_{1}(n-1)^{m}+^{n}C_{2}(n-2)^{m}– ………….. + (-1)^{n-1n}C_{n-1}1^{m}if m ≥ n.

For the composition of functions f and g be two functions :

- fog ≠ gof
- If f and g both are one-one function then fog is also one-one.
- If f and g both are onto function then fog is also onto.
- If f and fog both are one-one function then g is also one-one.
- If f and fog both are onto function then it is not necessary that g is also onto.
- (fog)
^{-1}= g^{-1}o f^{-1} - f
^{-1}o f = f^{-1}(f(a)) = f^{-1}(b) = a - fof
^{-1 }= f(f^{-1}(b)) = f(a) = b

### Sample Questions:

**Ques 1: Show that the function f : R ⇢ R, given by f(x) = 2x, is one-one and onto.**

**Sol: For one-one:**

Let a, b ∈ R such that f(a) = f(b) then, f(a) = f(b) ⇒ 2a = 2b ⇒ a = bTherefore, f: R ⇢ R is one-one.

**For onto:**

Let p be any real number in R (co-domain). f(x) = p ⇒ 2x = p ⇒ x = p/2 p/2 ∈ R for p ∈ R such that f(p/2) = 2(p/2) = p For each p∈ R (codomain) there exists x = p/2 ∈ R (domain) such that f(x) = y For each element in codomain has its pre-image in domain.So, f: R ⇢ R is onto.Since f: R⇢R is both one-one and onto.f : R⇢R is one-one correspondent (bijective function).

**Ques 2: Let f : R ⇢ R ; f(x) = cos x and g : R ⇢ R ; g(x) = x ^{3} . Find fog and gof.**

**Sol: **Since the range of f is a subset of the domain of g and the range of g is a subset of the domain of f. So, fog and gof both exist.

gof (x)= g(f(x)) = g(cos x) = (cos x)^{3}= cos^{3}xfog (x)= f(g(x)) = f(x^{3}) = cos x^{3 }

**Ques 3: If f : Q ⇢ Q is given by f(x) = x ^{2} , then find f^{-1}(16).**

**Sol: **

Let f^{-1}(16) = x f(x) = 16 x^{2}= 16 x = ± 4 f^{-1}(16) = {-4, 4}

**Ques 4 :- If f : R ⇢ R; f(x) = 2x + 7 is a bijective function then, find the inverse of f.**

**Sol: **Let x ∈ R (domain), y ∈ R (codomain) such that f(a) = b

f(x) = y ⇒ 2x + 7 = y ⇒ x = (y -7)/2 ⇒ f^{-1}(y) = (y -7)/2Thus, f⇢^{-1 : R }R is defined as f^{-1(x) = (x -7)/2 for all x∈ R.}

**Ques 5: If f : A ⇢ B and |A| = 5 and |B| = 3 then find total number of functions.**

**Sol:** Total number of functions = 3^{5}** **= 243