Relations & Functions

Relations & Functions

BASIC CONCEPTS

Natural Numbers:

The numbers 1, 2, 3, 4, .... are called natural numbers, and their set is denoted by N.

Thus N = {1, 2, 3, 4, 5, ....}.


Integers:

The numbers ....-3, -2, -1, 0, 1, 2, 3.... are called integers and the set is denoted by I or Z. integers cane be positive, negative, non positive and non negative.


Rational Numbers:
All numbers of the form p/q where p and q are integers and q 0, are called rational numbers and their set is denoted by Q.


Irrational Numbers:

There are numbers which cannot be expressed in the p/q form. These numbers are called irrational numbers and their set is denoted by Q^c (i.e. complementary set of Q)


Real Numbers:
The complete set of rational and irrational numbers is the set of real numbers and is denoted by R. Thus R = QQ^c.

It may be noted that . Real numbers can also be expressed in terms of position of a point on the real line. The real line is the number line wherein

the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa.





Real line

All the numbers defined so far follow the order property i.e. if there are two numbers a and b then either a< b or a = b or a > b.

Intervals:

Intervals are basically subsets of R. If there are two numbers a, b R such that a < b, we can define four types of intervals as follows:

Open Interval i.e. end points are not included. Closed Interval i.e. end points are also included. This is possible only when both a and b are finite.

Open-closed Interval: (a, b] = { x: a < x b}

Closed -open Interval: [a, b) = { x: a x < b}

The infinite intervals are defined as follows:

(a, ) ={ x: x > a}

(a, ) = { x: x a}

(-, b) = {x: x< b)

(-, b] = { x: x b}

Inequalities:

a b either a < b or a = b


a < b and b < c a < c


a < b a + c < b + c c R


a < b -a > -b i.e. inequality sign reverses if both sides are multiplied by a negative number


a < b and c < d a + c < b + d and a - d < b - c


a < b ma < mb if m > 0 and ma > mb if m< 0


0 < a < b a^r < b^r if r > 0 and a^r > b^r if r < 0


2 a > 0 and equality holds for a = 1


-2 a < 0 and equality holds for a = -1

Absolute Value:

Let x R. Then the magnitude of x is called it's absolute value and is, in general, denoted by |x|.Thus |x|can be defined as,



Note that x = 0 can be included either with positive values of x or with negative values of x. As we know that all real numbers can be plotted on the real number

line, |x|represents the distance of number 'x' from the origin, measured along the number-line. Thus |x|0.Secondly, any point 'x' lying on the real number line

will have it's coordinates as (x, 0). Thus, it's distance from the origin is |x| =.

Basic Properties of |x|:




> a x > a or x < -a if a R⁺, and x R if a R^-


< a - a < x < a if a R⁺ and no solution if a R^- {0}







The last two properties can be put in one compact form namely,








Greatest Integer and Fractional Part

Let x R. Then [x] denotes the greatest integer less than or equal to x and {x} denotes the fractional part of x and is given by {x} = x - [x]. Note that 0

{x} < 1.It is obvious that if x is an integer, then [x] = x.

Basic Properties of Greatest Integer and Fractional Part:

[[x]] = [x], [{x}] = 0, {[x]} = 0


x - 1 < [x] x, 0 {x} < 1


[n + x] = n + [x] where n I







.


Hence [x] + [y] [x + y] [x] + [y] +1




Relations:

Let A and B be two sets. A relation R from the set A to set B is a subset of the cartesian product . Further, if , then we say that x is R-related to y

and write this relation as x R y. Hence.

We notice that in every ordered pair of R, the second element is the cube of the first element i.e. the element of the ordered pairs of R have a common relation-

ship which is "cube".

Here the first element in each of the ordered pair is greater than the second element. Hence the relationship is "greater than". Obviously, from the definition, x R y

and y R x are not the same, since and are different.

Domain and Range of a relation: Let R be a relation defined from a set A to a set B, i.e.. Then the set of all first elements of the ordered pairs in R is

called the domain of R. The set of all second elements of the ordered pairs in R is called the range of R. That is, D = domain of or

,

= range of or .

Clearly and.

Inverse relation:

Let R be a relation from a set A to a set B. Then, the inverse relation of R, denoted by , is a relation defined by ,


Types of relation in a set A are:

(i) Identity relation:

A relation R in the set A defined by or is called the identity relation.


(ii) Void relation:

A relation R in the set A is void relation if.

(iii) Universal relation:

A relation R in the set A defined as is called the universal relation in the set.

(iv) Reflexive relation:

A relation R, in a set A, is called a reflexive relation if for all or for all.

(v) Symmetric relation:

A relation R, in a set A, is called a symmetric relation if or.


(vi) Transitive relation:

A relation R, in a set A, is called a transitive relation if and or and .

(vii) Equivalence relation:

A relation R, in a set A is an equivalence relation if R is reflexive, symmetric and transitive i.e. for all, , and.


Functions

Function is a special case of a relation, since a relation may relate an element of A to more than one elements in B. A function from A to B is usually denoted by

the symbols f, g etc. and we write. We also say that "f is a mapping from A to B".

The set A is called the domain of the function f and B is called the co domain of the function f.



(i) Graph of a function:

Letbe a function. Then the subset of is called the graph of the function f. We know that a relation R from the set A to B is a

subset of. The relation R is a function from the set A to the set B, if every element of A is the first element of exactly one ordered pair of R. The function f

from the set A to the set B is usually written explicitly;




Definitions of Function, Domain, Co-domain and Range:

Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. "A function from X to Y is a rule or

correspondence that assigns to each element of set X, one and only one element of set Y". Let the correspondence be 'f'. Then mathematically we write f:X

Y where y = f(x), x X and y Y. We say that 'y' is the image of 'x' under 'f'(or x is the pre image of y). Accordingly

A mapping f: X Y is said to be a function if each element of X has it's image in the set Y. It is possible that a few elements in Y are present which are
not the images of any element in the set X.

Every element of X should have one and only one image. That means it is impossible to have more than one images for a specific element of X. Functions

can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y)

The set 'X' is called the domain of 'f'.

The set 'Y' is called the co-domain of 'f'.

Set of images of different elements of the set X is called the range of 'f'. It is obvious that range could be a subset of the co-domain as we may have some

elements in the co-domain which are not the images of any element of X (of course, these elements of the co-domain will not be included in the range).

Range is also called domain of variation.


If R is the set of real numbers and X and Y are subsets of R, then the function

f (x) is called a real-valued function or a real function.

Domain of a function 'f' is normally represented as Domain (f). Range is represented as Range (f). We are primarily interested in functions whose domain and

ranges are (sub) sets of real numbers.


Graphs of Some Elementary Functions

FUNCTION	DOMAIN, RANGE AND DEFINITION		GRAPH
1. The constant function The graph of y = f (x) is a line parallel to the x-axis. If c > 0, the line is above the x-axis. If c = 0, the line coincides with the x-axis If c < 0, the line is below the x-axis.	A function f : A B, A, B R, is said to be a constant function if there exists a real number c such that f (x) = c for all x A. Domain : A Range : {c}
2. The identity function The graph of y = f (x) = x is a straight line with slope 1. The line passes through the origin	The function from R R that associates to each x R the same x, is called the identity function. More precisely f (x) = x Domain : R Range : R
3. The absolute value (or modulus) function. The graph of y= f (x) = |x| is symmetric about the y-axis. It passes through the origin and remains above the x-axis.	The function f : R R+ + {0} defined by f (x) = |x| = x if x 0 - x if x < 0 is called the absolute value function Domain : R Range : [0, ).
4. The exponential function The graph of y = ex lies above the x-axis As we move from left to right, the value of y increases (i.e. the graph rises) It meets the y-axis at (0, 1) For negative values of x, the graph approaches the x-axis, but never meets it.	If a is positive real number and a 1, then the function defined by f (x) = ax, x R is called an exponential function to the base a. If a = e, the exponential function takes the form f (x) = ex, x R. Domain : R Range : (0, )
5. The Logarithmic Function The graph of y = ln x is on the right side of the y-axis y increases as x increases The graph meets the x-axis at (1, 0) For values of x < 1, the graph approaches the y-axis but never meets it.	If a > 0 and a 1, then the function loga x : R+ R, given by y = loga x, if and only if ay = x, is called the logarithmic function. For a = e, we call it the natural logarithmic function i.e. f : R+ R, y = loge x = ln (x) iff ey = x. Domain = R+ (0, ) Range = R
6. The Greatest Integer Function [x] denotes the greatest integer less than or equal to x. In particular x - 1 < [x] x. It is also called a step function. The graph of y = [x] consists of infinitely many broken pieces, in the first and the third quadrant, each piece parallel to the x-axis.	The function f : R Z defined by f(x) = [x], x R is called the greatest integer function Domain : R Range : Z.
7. The fractional part function If x = [x] + {x}, 0 {x} < 1, then {x} is the fractional part of x. The graph of y = {x} consists of infinitely many broken pieces, each piece parallel to the line y = x, varying from y = 0 to y = 1.	The function f : R [0, 1) defined by f(x) = {x} is called the fractional part function. Domain : R Range: [0, 1)
8. Polynomial Function	f(x) = a0xn + a1xn - 1 + .... +an - 1x + an where a0, a1,....,an are real numbers, a0 0. Domain : R Range : Depends on the polynomial representing the function.
9. Rational Function	f(x) = , where p(x) and q(x) are polynomials in x. Domain is R - {x : q(x) = 0}, Range depends on the function.

1. Constant function:

Let be a function from. f is said to be a constant function if there exists an element such that, for all.




2. Identity function:

Let A be a non-empty set. If the mapping is such that each element of the set A is mapped onto itself, then f is said to be an identity function. The

identity function is a bijective function.


3. Trigonometric or Circular Functions:

The domains and ranges of these functions are:

Function	Domain	Range
sin x	R	[- 1, 1]
cos x	R	[- 1, 1]
tan x	R -	R
cot x	R -	R
sec x	R -	(- , - 1] [1, )
cosec x	R -	(- , - 1] [1, )

The graphs of the trigonometric functions are given below:




Algebra of Functions:

(i) Let f and g be two functions defined from A to B. Then are equal if.

If the function f and g are equal, then the subsets, graph of f and graph of g, of are equal.

(ii) Given functions f : D R and g : D R, we define functions f + g, f - g, gf and f/g as follows:

f + g : D R is a function defined by (f + g)(x) = f(x) + g(x) for all x D


f - g : D R is a function defined by (f - g) (x) = f(x) - g(x) for all x D


fg : D R is a function defined by (f g) (x) = f(x) g(x) for all x D


:CR is a function defined by (f/g) (x) = ,where C={xD:g(x) 0}. i.e. x D - {x : g (x) = 0}.


If k is any scalar, then kf is a function from DR defined by (kg)(x) = kf(x), xD.

Note: The sum f + g, the difference f - g, the product fg and the quotient f/g are defined only when f and g are real functions having the same domain. In case f and

g, have different domains, these operations are defined for those values of x which are common to the domains of both f and g i.e. x to the intersection of the

domains of f and g.


Composite Functions In the figure shown below correspondence between the elements of A and C is called the composite function of f and g and is denoted by

gof i.e. the composite mapping is defined by: such that for all.

The function h, defined above, is called the composition of f and g. Domain of gof = {x : x in domain f, f(x) in domain g}.




Note : (i) The range of f is the domain of g. A is the domain of gof and C is its range.

In general, composite function of two functions is not commutative i.e..

In particular if f is a bijection of A onto itself then,

f^-1 of = f of^-1 = I, where I is the identity function.

(ii)The function fog is not always defined,


Even and Odd Functions

Let f : D R be a real function such that - x D x D. Then f is called an even function if

f(-x) = f(x) for every x D and an odd function if f(- x) = - f(x) for every x D.

The domain and range of f depend on the definition of the function f. Graph of even function is symmetrical in I and II (or III and IV) quadrants whereas graph of

odd function is always symmetrical in diagonally opposite quadrants.

For every function f(x), the function f(x)+ f(-x) is even and the function f (x)- f (- x) is odd.

Every function f(x) can be represented as the sum of an even function and an odd function as f (x) = [f (x) + f (- x)] + [f (x) - f (- x)].

Periodic Functions

If a function satisfies the condition f(x + p) = f (x), for all x X, where p is the smallest positive real number, independent of x, then f (x) is said to be a

periodic function and p is called the period of the periodic function.


Rules for finding the period of the periodic functions:

(i) If p, independent of x, is the solution of the equation f (x + p) = f (x), then f (x) is periodic with period p.

(ii) If f (x) is a periodic function with period p, then f (x + np) = f (x), n = 1, 2, ........

(iii) If f(x) is periodic with period p, then a f(x) + b, where a, b R (a 0) is also a periodic function with period p.

(iv) If f(x) is periodic with period p, then f(ax + b), where a R-{0} and b R, is also periodic with period .

(v) If f (x) is periodic with period p and g (x) is periodic with period q, and r is the LCM of p and q (if it exists), then r is the period of f (x) + g (x) provided there

does not exist a constant c such that f (c + x) = g (x) and g (c + x) = f (x).


One-to-one functions:

Let be a function from the set A to the set B. Then f is said to be one-to-one function if the images of distinct elements of A are distinct elements of B

i.e. if.



A one-to-one function is also called an injective function.

A function which is not one-one is called a many-one function.


Methods to Find One-One or Many-One:

(i) If x₁ x₂ f(x₁) f(x₂) x₁, x₂ in the domain, then f(x) is one-one.

(ii) If f(x₁) = f(x₂) x₁ = x₂, then f(x) is one-one.

(iii) Any function f(x), which is entirely increasing or decreasing in the whole of a domain, is one-one.

(iv) Any continuous function f(x), which has at least one local maximum or local minimum, is many-one.

(v)If any line parallel to the x-axis cuts the graph of the function at the most at one point, then the function is one-one and if there exists a line which is parallel to

the x-axis and cuts the graph of the function in at least two points, then the function is many-one.

Remark: Any function will be either one - one or many - one.

Onto and Into functions:

Let be a function from the set A to set B. Then, f is said to be an onto function (onto mapping) if every element of B is image of at least one element of A.




If on the other hand, range of f is a subset of B, then f is said to be an into function. In other words, if each element in the co-domain have at least one preimage in

the domain,

Note: Onto function is called surjective function and a function which is both one-one and onto is called bijective function.

Methods to find Onto or Into function:

(i) If range = co-domain, then f(x) is onto and if range is a proper subset of the co-domain, then f(x) is into.

(ii) Solve f(x) = y by taking x as a function of y i.e. g(y) (say).

Remark: Any polynomial function f : R R is onto if degree of f is odd and into if degree of f is even.

One-to-one and onto functions:
Let be a function from the set A to the set B. f is said to be one-to-one and onto if it is both one-to-one and onto.
A one-to-one and onto function is also called a bijective function.




Inverse Function

Let be a function from the set A to the set B. Let. The inverse image of the element is the set of all elements of A whose image under the

mapping f is b. If is one-to-one and onto, then the inverse image is called the inverse function of f.

Since f is onto, for is non-empty. In fact is a singleton set. Hence, for every, there exists a unique element such that.

This correspondence between the elements of B and A is called the inverse function of f and is denoted by. Hence if and only if i.e.

is defined only when f is a bijective function. The function is also a bijective function.

Further, if g is the inverse of f, then f is the inverse of g and the two functions f and g are said to be the inverses of each other. For the inverse of a function to exist,

the function must be one-one and onto.


Method of finding Inverse of a Function:

If f ^-1 is the inverse of f, then fof ^-1 = f ^-1 o f = I, where I is an identity function.

fof ^-1 = I (fof ^-1(x)) = I (x) = x.

Operate f on f ^-1 (x), and get an equation in f ^-1 (x) and x.Solve it to get f ^-1 (x).

The domain of f^-1 is the range of f (x) and the range of f^-1 is the domain of f (x).

Note: A function and its inverse are always symmetric with respect to the line y = x. To draw the graph of f^-1, we simply interchange the roles of x and y.

We give below some standard functions along with their inverse functions:

Function	Inverse Function
1. f : [0, ) [0, ) defined by f(x) = x2	f -1 : [0, ) [0, ) defined by f -1 (x) =
2. f : [-1, 1] defined by f(x) = sinx	f -1 [-1,1] defined by f -1 (x) = sin-1x
3. f : [0, ] [-1,1] defined by f(x) = cos x	f -1 : [-1, 1] [0, ] defined by f -1 (x) = cos-1x
4. f : (-, ) defined by f(x) = tan x	f -1 : (-, ) defined by f -1 (x) = tan‑1x
5. f : (0, ) (-, ) defined by f(x) = cot x	f -1 : (-, ) (0, ) defined by f -1 (x) = cot-1x
6. (-,-1]U[1, ) defined by f(x) = sec x	f -1 : (-, -1] U [1, ) defined by f -1 (x) = sec-1x
7. defined by f(x) = cosec x	f -1 : (-, -1] U [1, ) defined by f -1 (x) = cosec-1x
8. f : R R+ defined by f(x) = ex	f -1(x) : R+ - R defined by f -1 (x) = ln x.

Binary Operations:

Let A be a non-empty set. Then a function is called a unitary operation on A. A function is called a binary operation on A. The binary

operation is usually denoted by "o" or "*". The image of (a, b) (A × A) under the binary operation* is denoted by a* b. Similarly, a function

is called an n-ary operation on A.


Laws of binary compositions:

Let A be a non-empty set and '*' be a binary operation defined on A.

Commutative composition. The binary operation '*' is said to be commutative if

a*b = b*a for a, b A.


Associative composition. The binary operation '*' is said to be associative if

(a*b) *c = a* (b*c) for a, b, c A.


Identity element. An element e A is said to be an identity element for the binary operation if a*e = a = e*a for a A.

For binary operation of addition in R, 0 (zero) is the identity element. For multiplication, the identity element is 1.

For a binary operation, if the identity element exists, then it is unique.