Ordered pair

A pair having one element as the first and the other as the second is called an ordered pair. The ordered pair having ‘a’ as the first element and ‘b’ as the second element is denoted by (a, b).

Cartesian product

Let A and B be two non-empty sets. The Cartesian product of set A and B is denoted by A ´ B and is defined by A ´ B = {(a, b): a Î A, b Î B}

If A = {1, 2} and B = {a, b}, then A ´ B = {(1, a), (1, b), (2, a), (2, b)}.

Relation: Any subset of a Cartesian product A ´ B of two non- empty sets A and B is called a relation.

Domain and Range of relation: The domain of relation R is the set of all first members of the ordered pair and it is denoted by Dom (R).

The range of the relation R is the set of all second members of the ordered pair and it is denoted by Ran (R).

Inverse of the relation: If R = {(a, b) : a ÎA, b Î B} Ì A ´ B is the relation of set A and B, the relation R-1 defined by R-1 = {(b, a) : b Î B, a Î A} Ì B ´ A is called the inverse relation of R.

Function

Let A and B be two non-empty sets, then the relation from set A to set B is known as the function from set A to set B when each element of set A associate with a unique element of B. A function from set A to set B is denoted by f: A ® B.

Types of function

·               One-one or injective function

            A function f: A ® B is said to be one-one or injective if distinct elements in A have distinct images in B.

            For x, y Î A, x ¹ y Þ f(x) ¹ f(y)

            or, f(x) = f(y) Þ x = y

·               Onto or subjective function

            A function f: A ® B is said to be onto or subjective, if every element of B is the image of at least one element of A, i.e. f(A) = B

·               One-one onto or objective function

            Let f: A ® B is both one-one and onto is called objective function.

Inverse function

Let f : A ® B is a one-one and onto function, the function from B to A such that every element of B associates with unique element of A, then the function defined from B to A is known inverse function of f and is denoted by f-1.

In short, if f : A ® B is bijective, then these exists a function f-1: B®A called inverse function of f

Composite function

If f: A ® B and g: B®C be any two functions, then the composite function of f and g is the function g o f: A ®C defined by the equation (g o f) (x) = g[f(x)]

Some Algebraic Function

·         The identity function

    Let A be any non-empty set. The function f: A ® A defined by y = f(x) = x for x Î A, is called the identity function.

The graph of the identity function is a straight line passing through origin and making angle of 45o with x-axis.

·         Constant function

        Let A be any set and B = {c}. Then, the function f : A® B defined by

        y = f(x) = c for x Î A, is called a constant function.

·         The linear function

        Let A and B be any two sets. Then the function f : A ® B defined by            y = f(x) = mx + c for x Î A, where m and c are constants is called a linear function.

·         Quadratic function

  Let A and B be two sets. Then the function f : A ® B defined by

y = f(x) = ax2 + bx + c for x Î A, where a, b, c are constants, is called a quadratic function.

·      Cubic function

Let A and B be any two sets. Then, the function f: A ® B defined by

    y = f(x) = ax3 + bx2 + cx + d, where a, b, c, d are constant, is called a cubic function.

·         Polynomial function

                  Let A and B be two sets. Then a function f: A® B defined by

              y = f(x) = anxn + an- 1xn-1 + …..a1x1 + a0 for x Î A where a0, a1,…,an are constants is called a polynomial function.

Solved Problems 

          Relations


Functions and Graphs