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Test 11 of 39

# Functions

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Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).

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 to B, we write  f : A → B (read as f is a mapping from to B).
3. If f associates x ∈ A to y ∈ B then we say that y is the image of the element under the map (or function) f and we write (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 (x). Thus f is onto if range of = 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((x)), for all x ∈ A

# Inverse of a function

1. If and g be two functions satisfying (g(x)) = x, for every x in the domain of g, and g(f(x)) = x for every 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 yf(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 is a function   ∗: A × A  . We denote ∗ (a, b) by ∗ b.
• It follows from the definition of a function that a binary operation on a set 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 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 and if there exists an element e ∈ A, such that a∗e = a = e∗a,  for all a ∈ A, then is called an identity element for the binary operation ′∗′  on set and it is unique.
• (iv) Inverse of an element: Let ′∗′  be a binary operation on a set A, and let be the identity element in 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 is called the inverse of an element a, and it is unique.
• Note: The inverse of a element is generally denoted by a-1