DEFINITIONS & RESULTS
GENERAL DEFINITION :
If to every value (considered as real unless other-wise stated) of a variable x, which belongs to some collection
or a dependent variable
(Set) A, there corresponds one and only one finite value of the quantity valued) of x
belonging to some set A there corresponds one or several values of the variable y, then
as the meaning of a single valued function, if not otherwise stated .
y is called a multiple valued function of x defined on A .Conventionally the word "FUNCTION” is used only
Every function from A
conditions .
® B satisfies the following
(a)
f Ì A x B
(b)
" a Î AÞ (a, f(a)) Î f and
(c)
(a, b) Î f & (a, c) Î f Þ b = c2. DOMAIN, CO-DOMAIN & RANGE OF A FUNCTION :
Let f : A
of f & the set B is known as co-domain of f . The set of all f images of elements of A is known as the range of f .
Thus : Domain of f = {a
Range of f = {f(a) | a
It should be noted that range is a subset of co-domain .Sometimes if only f (x) is given then domain is set of
those values of ' x ' for which f (x) exists or is defined .
To find the range of a function, there isn't any particular approach, but student will find one of these approaches
useful .
® B, then the set A is known as the domain| a Î A, (a, f(a)) Î f}Î A, f(a) Î B}
(i)
if possible ' x ' as a function of
' y ' i.e. x = g (y) . Find the domain of ' g ' . This will
become range of ' f ' .
When a function is given in the form y = f (x), express
(ii)
function , then range of ' f ' will be union of [ Min
Max
piece-wise continuous .
If y = f (x) is a continuous or piece-wise continuousm f (x),m f (x) ] in all such intervals where f (x) is continuous/
3. C
LASSIFICATION OF FUNCTIONS :
Functions can be classified into two categories :
(i) One - One Function (Injective mapping) or Many - one function :
® B is said to be a one-one function or injective mapping if different elements of A have
different f images in B . Thus for x
f(x
x
1, x2 Î A & f(x1) ,2) Î B , f(x1) = f(x2) Û x1 = x2 or1 ¹ x2 Û f(x1) ¹ f(x2) .
Note:(a)
Any function which is entirely increasing or decreasing in whole domain, then f(x) is one-one .
(b)
one-one .
If any line parallel to x-axis cuts the graph of the function atmost at one point, then the function is
Many - one function :
A function f : A
image in B . Thus f : A
x
® B is said to be a many one Function if two or more elements of A have the same f® B is many one if for ;1, x2 ÎA , f(x1) = f(x2) but x1 ¹ x2 .
Note: (a)
In other words, if there is even a single line parallel to x-axis cuts the graph of the function atleast at two points,
then f is many-one .
(b)
If a function is one-one, it cannot be many-one and vice versa .
(c)
All functions can be categorized as one-one or manyone
(ii) Onto function (Surjective mapping) or into function:
If the function f : A
we say that f is a function of A 'onto' B . Thus f : A
f (a) = b .
® B is such that each element in B (co-domain) must have atleast one pre-image in A, then® B is surjective iff " b Î B, $ some a Î A such that
Note that :
Into function :
If f : A
domain, then f(x) is into .
® B is such that there exists atleast one element in co-domain which is not the image of any element in
Note that :
Thus a function can be one of these four types :
if a function is onto, it cannot be into and viceversa
(a)
one-one onto (injective & surjective)
(b)
one-one into (injective but not surjective)
(c)
many-one onto (surjective but not injective)
(d)
many-one into (neither surjective nor injective)
Note : (a)
named as invertible, non singular or biuniform functions.
If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also
(b)
If a set A contains n distinct elements then the number of different functions defined from A® A is nn
& out of it n ! are one one .
5. Important Types Of Functions :
(i) Polynomial Function :
If a function f is defined by f (x) = a
..., a
0 xn + a1 xn-1 + a2 xn-2 + ... + an-1 x + an where n is a non negative integer and a0, a1, a2,n are real numbers and a0 ¹ 0, then f is called a polynomial function of degree n .
N
OTE : (a) A polynomial of degree one with no constant term is called an odd linear function . i.e. f(x) = ax , a ¹ 0
(b)
There are two polynomial functions , satisfying the relation ; f(x).f(1/x) = f(x) + f(1/x) . They are :
(i)
f(x) = xn + 1 &
(ii)
f(x) = 1 - xn , where n is a positive integer.
(ii) Algebraic Function :
y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form , P
....... + P
e.g. y =
Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called
n-1 (x) y + Pn (x) = 0 Where n is a positive integer and P0 (x), P1 (x) ........... are Polynomials in x.| x | is an algebraic function, since it satisfies the equation y² - x² = 0.
T
RANSCEDENTAL FUNCTION .
(iv) Exponential Function :
A function f(x) = a
exponential function is called the logarithmic function .
i.e. g(x) = log
x = exlna(a > 0 , a ¹ 1, x Î R) is called an exponential function . The inverse of thea x .
(vii) Greatest Integer Or Step Up Function :
The function y = f (x) = [x] is called the greatest integer
function where [x] denotes the greatest integer less than
or equal to x .
- 1 < x < 0 ; [x] = - 1 0 < x < 1 ; [x] = 0
1 < x < 2 ; [x] = 1 2 < x < 3 ; [x] = 2
and so on .
Note that for :
Properties of greatest integer function :
(a)
x - 1 < [x] < x , 0 < x - [x] < 1
[x] < x < [x] + 1 and
(b)
[x + m] = [x] + m if m is an integer .
(c)
[x] + [y] < [x + y] £ [x] + [y] + 1
(d)
[x] + [- x] = 0 if x is an integer = - 1 otherwise .
(viii) Fractional Part Function :
It is defined as : g (x) = {x} = x - [x] .
e.g. the fractional part of the no. 2.1 is 2.1- 2 = 0.1 and the
fractional part of - 3.7 is 0.3 . The period of this function is 1
(ix) Identity function :
The function f : A
observe that identity function is a bijection .
® A defined by f(x) = x " x Î A is called the identity of A and is denoted by IA . It is easy to
(x) Constant function :
A function f : A
f : A
singleton and a constant function may be one-one or many -one, onto .
® B is said to be a constant function if every element of A has the same f image in B . Thus® B ; f(x) = c , " x Î A , c Î B is a constant function. Note that the range of a constant function is a
6. Homogeneous Functions :
An integral function is said to be homogeneous with respect to any set of variables when each of its terms
is of the same degree with respect to those variables.
For example 5 x
2 + 3 y2 - xy is homogeneous in x & y .
7. Bounded Function :
A function is said to be bounded if | f(x) | < M , where M is a finite quantity .
8. Implicit & Explicit Function :
A function defined by an equation not solved for the dependent variable is called an
eg. the equation x
it is called an
IMPLICIT FUNCTION . For3 + y3 = 1 defines y as an implicit function. If y has been expressed in terms of x alone thenEXPLICIT FUNCTION .
9. Odd & Even Functions :
If f (-x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function.
e.g. f (x) = cos x ; g (x) = x² + 3 .
If f (-x) = -f (x) for all x in the domain of ‘f’ then f is said to be an odd function.
e.g. f (x) = sin x ; g (x) = x
(a)
3 + x . NOTE :f (x) - f (-x) = 0 Þ f (x) is even & f (x) + f (-x) = 0 Þ f (x) is odd .
(b)
A function may neither be odd nor even .
(c)
Inverse of an even function is not defined .
(d)
Every even function is symmetric about the y-axis & every odd function is symmetric about the origin .
(e)
Every function can be expressed as the sum of an even & an odd function.
(f)
The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0 .
(g)
If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd .
10. Periodic Function :
A function f(x) is called periodic if there exists a + ve number T (T > 0) called the period of the function such
that f (x + T) = f(x), for all values of x within the domain of x .
e.g. The function sin x & cos x both are periodic over 2
p & tan x is periodic over p .
N
(a)
OTE :f (T) = f (0) = f (-T) , where ‘T’ is the period .
(b)
Inverse of a periodic function does not exist .
(c)
Every constant function is always periodic, with no fundamental period .
(d)
f (x) + g (x) must have a period T .
e.g. f (x) = | sinx | + | cosx | .
If f (x) has a period T & g (x) also has a period T then it does not mean that
(f)
If f(x) has a period T then f (ax + b) has a period T/ IaI
11. Composite Functions :
Let f : A
® B & g : B ® C be two functions . Then the function gof : A ® C defined by (gof) (x) = g (f(x)) " x
Î
A is called the composite of the two functions f & g .
Thus the image of every x
the g-image of the f-image of x.
Note that gof is defined only if
element of the domain of g so that we can take its gimage
. Hence for the product gof of two functions f & g,
Î A under the function gof is" x Î A, f(x) is an
Properties Of Composite Functions :
(i)
The composite of functions is not commutative i.e. gof ¹ fog .
(ii)
defined, then fo (goh) = (fog) oh .
The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are
(iii)
The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection .
12. Inverse Of A Function :
Let f : A
g : B
y
® B be a one-one & onto function, then their exists a unique function® A such that f(x) = y Û g(y) = x, " x Î A &Î B . Then g is said to be inverse of f . Thus g = f-1 : B ® A = {(f(x), x) | (x, f(x)) Îf} .
Properties Of Inverse Function :
(i)
The inverse of a bijection is unique .
(ii)
functions on the sets A & B respectively.
If f : A ® B is a bijection & g : B ® A is the inverse of f, then fog = IB & gof = IA , where IA & IB are identity(iii) The inverse of a bijection is also a bijection .
(iv)
If f & g are two bijections f : A® B , g : B ® C then the inverse of gof exists and (gof)-1 = f-1 o g-1 .
(v)
Inverse of an even function is not defined .
13. Equal or Identical Function :
Two functions f & g are said to be equal if :
(i)
The domain of f = the domain of g .
(ii)
(iii)
f(x) = g(x) , for every x belonging to their common
14. General :
The range of f = the range of g and domain .
If x, y are independent variables, then :
(i)
f(xy) = f(x) + f(y) Þ f(x) = k ln x or f(x) = 0 .
(ii)
f(xy) = f(x) . f(y) Þ f(x) = xn , n Î R
(iii)
f(x + y) = f(x) . f(y) Þ f(x) = akx .
(iv)
the range of f must be a subset of the domain of g .
f(x + y) = f(x) = f(y) Þ f(x) = kx, where k is a constant.
If to every value of
x
Any continuous function which has atleast one local maximum or local minimum, then f(x) is many-one . A function f : A
