Inverse Trigonometric Functions

SETS, RELATIONS & FUNCTIONS

2.2.3 Inverse of a relation : Let A, B be two sets and let R
RELATION & FUNCTION­I be a relation from a set A to set B. Then the inverse
–1
of R, denoted by R , is a relation from B to A and is
1. INTRODUCTION defined by
–1
In this chapter, we will learn how to create a relation between R = {(b, a) : (a, b)  R}
–1
two sets by linking pairs of objects from two sets. We will Clearly, (a, b)  R œ (b, a)  R
–1 –1
learn how a relation qualifies for being a function. Finally, Also, Dom (R) = Range (R ) and Range (R) = Dom (R ).
we will see kinds of function, some standard functions etc.
3. FUNCTIONS
2. RELATIONS
3.1 Definition

2.1 Cartesian product of sets A relation ‘f’ from a set A to set B is said to be a function if
every element of set A has one and only one image in set B.
Definition : Given two non-empty sets P & Q. The cartesian
product P × Q is the set of all ordered pairs of elements from Notations
P & Q i.e.
P × Q = {(p, q); p  P; q  Q}

2.2 Relations

2.2.1 Definition : Let A & B be two non-empty sets. Then
any subset ‘R’ of A × B is a relation from A to B.
If (a, b)  R, then we write it as a R b which is read as
a is related to b’ by the relation R’, ‘b’ is also called
image of ‘a’ under R.
2.2.2 Domain and range of a relation : If R is a relation
from A to B, then the set of first elements in R is
called domain & the set of second elements in R is
called range of R. symbolically.
Domain of R = { x : (x, y)  R}
Range of R = { y : (x, y)  R}
The set B is called co-domain of relation R.
Note that range  co-domain.



3.2 Domain, Co-domain and Range of a function

Domain : When we define y = f (x) with a formula and the domain
is not stated explicitly, the domain is assumed to be the largest set
Total number of relations that can be defined from a set A of x–values for which the formula gives real y–values.
to a set B is the number of possible subsets of A × B. If
The domain of y = f (x) is the set of all real x for which f (x) is
n(A) = p and n(B) = q, then n(A × B) = pq and total defined (real).
pq
number of relations is 2 .

, SETS, RELATIONS & FUNCTIONS

Algo Check : Rules for finding Domain :

(i) Expression under even root (i.e. square root, fourth root etc.)
should be non–negative.
Two functions f & g are said to be equal iff
(ii) Denominator z 0.
1. Domain of f = Domain of g
(iii) logax is defined when x > 0, a > 0 and a z 1.
2. Co-domain of f = Co-domain of g
(iv) If domain of y = f (x) and y = g(x) are D1 and D2 respectively,
then the domain of f (x) ± g(x) or f (x) . g(x) is D1 ˆ D2. While 3. f(x) = g(x)  x  Domain.

f x 3.3 Kinds of Functions
domain of is D1 ˆ D 2 – {x: g(x) = 0}.
g x


Range : The set of all f -images of elements of A is known as the
range of f & denoted by f (A).

Range = f (A) = {f (x) : x  A};

f (A)ŽB {RangeŽCo-domain}.

Algo Check : Rule for finding range :

First of all find the domain of y = f (x)

(i) If domain  finite number of points

Ÿrange  set of corresponding f (x) values.

(ii) If domain  R or R – {some finite points}

Put y = f(x)

Then express x in terms of y. From this find y for x to be
defined. (i.e., find the values of y for which x exists).

(iii) If domain  a finite interval, find the least and greater value
for range using monotonocity.




1. Question of format :

§ Q L Q · Q o quadratic
¨y ; y ; y ¸
© Q Q L ¹ L o Linear
(a) One-to-One functions are also called Injective
Range is found out by cross-multiplying & creating functions.
a quadratic in ‘x’ & making D t 0 (as x  R) (b) Onto functions are also called Surjective
2. Questions to find range in which-the given (c) (one-to-one) & (onto) functions are also called
expression y = f(x) can be converted into x (or some Bijective Functions.
function of x) = expression in ‘y’.
Do this & apply method (ii).