INTRODUCTION TO FUNCTION | RELATION & FUNCTION

Function:

A Function is an ordered triple <∱,A,B> such that

1. A and B are sets and ∱ ⊆ A × B
2. For every x ∈A  there is some y ∈ B such that <x,y>∈∱
3.If <x,y>∈∱ and <x,z>∈∱ ,then y=z ;in other words the assignment is unique in the sense that an x ∈A is assigned  atmost one element of B.

• A is called Domain of ∱ and B is called Co-domain.

• Then we will usually write y=∱(x) ,and called " y is the image of x under ∱ "

• The set {y ∈ B :there is an x ∈A such that y=∱(x)}
  is called range of ∱ .

Types Of Function:

One to One Function:

A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a₁ ∈ A and b₁ ∈ B, f is defined as f:A→ B such that f (a₁) = f (b₁)

Many to One Function:

It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

Onto Function:

If there exists a function for which every element of set B there is (are) preimage(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.

One-One and Onto Function:

A function, f is One–One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.

Special Types Of Function:

Identity Function:

If A = B and f (x) = x for all x ∈ A, the f is called the identity function
on A .

Constant Function:

If f (x) = c for all x ∈ A, then f is called a constant function.

Polynomial Function:

A polynomial function is defined by y =a₀ + a₁x + a₂x² + … + a_nxⁿ, where n is a non-negative integer and a₀, a₁, a₂,…, _n ∈ R.

• Constant Function: If the degree is zero, the polynomial function is a constant function.
• Linear Function: The polynomial function with degree one. Such as y = x + 1
• Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function
• Cubic Function: A cubic polynomial function is a polynomial of degree three.

Rational Function:

A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x) ≠ 0.

Modulus Function:

The absolute value of any number, c is represented in the form of |c|. If any function f: R→ R is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,

f(x) = {x, if x ≥ 0; – x, if x < 0.

Signum Function:

Greatest Integer Function:

If a function f: R→ R is defined by f(x) = [x], x ∈ X. It round-off to the real number to the integer less than the number.It is also called as Floor Function.

For example: [5.12] = 5

Least Integer Function:

If a function f: R→ R is defined by f(x) = ⌈x⌉, x ∈ X. It round-off to the real number to the integer greater than the number. It is also called as Ceiling Function.

For example: [5.12] = 6