Function : Meaning

1. Let A and B be two non-empty sets, then a rule f which associates each element of A with a unique element of B is called a function or mapping from A to B.
2. If f is a mapping from A to B, we write  f : A → B (read as f is a mapping from A to B).
3. If f associates x ∈ A to y ∈ B then we say that y is the image of the element x under the map (or function) f and we write y = f (x). And, the element x is called the pre-image of y.

Types of Functions

• One-One Function (Injective) : A function f : A → B is said to be a one-one function or an injective if different elements of A have different images in B.

Thus, f : A → B is one-one. ⇔ a≠b  ⇒ f(a) ≠ f(b) for all a,b ∈ A or f(a) = f(b)⇔ a=b for all  a,b ∈ A

• Onto-Function (Surjective) : A function f : A → B is said to be an onto function or surjective if and only if each element of B is the image of some element of A i.e. for every element y ∈ B there exists some x ∈ A such that y = f (x). Thus f is onto if range of f = co-domain of f.
• One-one and onto Function (Bijective Function) : A function is a bijective if it is one-one as well as onto

Composition of Functions

• Suppose A, B and C be three non-empty sets and let  f : A → B,     g : B → C be two functions. since f is a function from A to B, therefore for each  x ∈ A there exists a unique element
• f(x) ∈ B  Again, since g is a function from B to C, therefore corresponding to f(x) ∈ B there exists a unique element  g(f(x)) ∈ C Thus, for each  fx ∈ A there exists a unique element g(f(x)) ∈ C .
• From above discussion it follows that when f and g are considered together they define a new function from A to C. This function is called the composition of f and g and is denoted by gof.
• i.e. gof(x) = g(f (x)), for all x ∈ A

Inverse of a function

1. If f and g be two functions satisfying f (g(x)) = x, for every x in the domain of g, and g(f(x)) = x for every x in the domain of f.
2. We say that f is the inverse of g and also, g is the inverse of f. We write  f ⁼ g⁻¹or g = f⁻¹
3. Tips: To find the inverse of f, write down the equation y= f(x) and then solve x as a function of y.
4. Thus the resulting equation is x = f⁻¹(y) ^.

Note :

1.  (i) If f is one-one function then f has inverse defined on its range
2. (ii) If f and g are one-one function then  (fog)⁻¹ = g⁻¹of⁻¹
3. (iii) If f is bijective then f⁻¹ is bijective and (f⁻¹)⁻¹  = f.

Binary Operation and Properties of Binary Operations

• Binary Operation : A binary operation ∗ on a set A is a function   ∗: A × A  . We denote ∗ (a, b) by a ∗ b.
• It follows from the definition of a function that a binary operation on a set A associates each ordered pair (a, b) ,
• (a,b) ∈ A × A to a unique element (a, b) in A.

Properties of Binary Operations

• (i) Commutativity: A binary operation ∗ on a set of A is said to be commutative binary operation, if  a ∗ b =b ∗ a   for all a,b ∈ A.

The binary operations addition (+) and multiplication (×) are commutative binary operation on z.

•  (ii) Associativity: A binary operation ∗: A × A → Ais said to be associative if

(a∗b)∗c =a∗(b∗c) ,  ∀ a,b,c ∈ A The binary operation of addition (+) and multiplication (×) are associative binary operation on z. However, the binary operation of subtraction is not associative binary operation.

• (iii) Identity Element: Let ′∗′ be a binary operation on a set A and if there exists an element e ∈ A, such that a∗e = a = e∗a,  for all a ∈ A, then e is called an identity element for the binary operation ′∗′  on set A and it is unique.
• (iv) Inverse of an element: Let ′∗′  be a binary operation on a set A, and let e be the identity element in A for the binary operation ∗ on S.Then, an element  a ∈ A is called an invertible element if there exists an element b ∈ A such that a×b = e = b∗a.
• The element b is called the inverse of an element a, and it is unique.
• Note: The inverse of a element is generally denoted by a^-1