Important Formulas: Functions With Examples | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


FORMULAE SHEET
Table: Domain and range of some standard functions-
Functions Domain Range
Polynomial function R R
Identity function x R R
Constant function K R (K)
Reciprocal function 
1
x
R
0
R
0
X
2
, x (modulus function)
R
{ }
R x
+
?
3
x ,x x
R R
Signum function 
x
x
R {-1,0,1}
X+ x
R
{ }
R x
+
?
x- x
R
{ }
R x
-
?
[x] (greatest integer function) R 1
x-{x} R [0,1]
x
(0, 8) [0,8]
a
x
(exponential function) R R
+
Log x(logarithmic function) R
+
R
RELATIONS	AND	FUNCTIONS
Page 2


FORMULAE SHEET
Table: Domain and range of some standard functions-
Functions Domain Range
Polynomial function R R
Identity function x R R
Constant function K R (K)
Reciprocal function 
1
x
R
0
R
0
X
2
, x (modulus function)
R
{ }
R x
+
?
3
x ,x x
R R
Signum function 
x
x
R {-1,0,1}
X+ x
R
{ }
R x
+
?
x- x
R
{ }
R x
-
?
[x] (greatest integer function) R 1
x-{x} R [0,1]
x
(0, 8) [0,8]
a
x
(exponential function) R R
+
Log x(logarithmic function) R
+
R
RELATIONS	AND	FUNCTIONS
Inverse Trigo Functions Domain Range
sin
–1
x (-1,1]
,
22
? ? -p p
? ?
? ?
cos
–1
x [-1,1]
[0, p ]
tan
–1
x R
,
22
? ? -p p
? ?
? ?
cot
–1
x R
(0, p )
sec
–1
x R-(-1,1)
[0, p ]-
2
?? p
??
??
cosec
–1
x R-(-1,1)
,
22
? ? -p p
? ?
? ?
-{0}
Inverse function: f
–1
 exists iff f is both one–one and onto.
f
-1
:B?A, f
-1
(b)=a ? f(a)=b
Even and odd function: A function is said to be
(a) Even function if f(x)=f(x) and
(b) Odd function if f(–x)= –f(x)
Properties of even & odd function:
(a) The graph of an even function is always symmetric about y-axis.
(b) The graph of an odd function is always symmetric about origin.
(c) Product of two even or odd function is an even function.
(d) Sum & difference of two even (odd) function is an even (odd) function.
(e) Product of an even or odd function is an odd function.
(f) Sum of even and odd function is neither even nor odd function.
(g) Zero function, i.e. f(x) = 0, is the only function which is both even and odd.
(h) If f(x) is an odd (even) function, then f
’
(x) is even (odd) function provided f(x) is differentiable on R.
(i)A given function can be expressed as sum of even and odd function.
i.e. 
( ) ( ) ( ) ( ) ( )
1 1
fx fx f x fx f x
2 2
? ? ? ?
= + - + --
? ? ? ?
=even function + odd function.
Increasing function: A function f(x) is an increasing function in the domain, D if the value of the function does not 
decrease by increasing the value of x.
Decreasing function: A function f(x) is a decreasing function in the domain, D if the value of function does 
not increase by increasing the value of x.
Page 3


FORMULAE SHEET
Table: Domain and range of some standard functions-
Functions Domain Range
Polynomial function R R
Identity function x R R
Constant function K R (K)
Reciprocal function 
1
x
R
0
R
0
X
2
, x (modulus function)
R
{ }
R x
+
?
3
x ,x x
R R
Signum function 
x
x
R {-1,0,1}
X+ x
R
{ }
R x
+
?
x- x
R
{ }
R x
-
?
[x] (greatest integer function) R 1
x-{x} R [0,1]
x
(0, 8) [0,8]
a
x
(exponential function) R R
+
Log x(logarithmic function) R
+
R
RELATIONS	AND	FUNCTIONS
Inverse Trigo Functions Domain Range
sin
–1
x (-1,1]
,
22
? ? -p p
? ?
? ?
cos
–1
x [-1,1]
[0, p ]
tan
–1
x R
,
22
? ? -p p
? ?
? ?
cot
–1
x R
(0, p )
sec
–1
x R-(-1,1)
[0, p ]-
2
?? p
??
??
cosec
–1
x R-(-1,1)
,
22
? ? -p p
? ?
? ?
-{0}
Inverse function: f
–1
 exists iff f is both one–one and onto.
f
-1
:B?A, f
-1
(b)=a ? f(a)=b
Even and odd function: A function is said to be
(a) Even function if f(x)=f(x) and
(b) Odd function if f(–x)= –f(x)
Properties of even & odd function:
(a) The graph of an even function is always symmetric about y-axis.
(b) The graph of an odd function is always symmetric about origin.
(c) Product of two even or odd function is an even function.
(d) Sum & difference of two even (odd) function is an even (odd) function.
(e) Product of an even or odd function is an odd function.
(f) Sum of even and odd function is neither even nor odd function.
(g) Zero function, i.e. f(x) = 0, is the only function which is both even and odd.
(h) If f(x) is an odd (even) function, then f
’
(x) is even (odd) function provided f(x) is differentiable on R.
(i)A given function can be expressed as sum of even and odd function.
i.e. 
( ) ( ) ( ) ( ) ( )
1 1
fx fx f x fx f x
2 2
? ? ? ?
= + - + --
? ? ? ?
=even function + odd function.
Increasing function: A function f(x) is an increasing function in the domain, D if the value of the function does not 
decrease by increasing the value of x.
Decreasing function: A function f(x) is a decreasing function in the domain, D if the value of function does 
not increase by increasing the value of x.
Periodic function: Function f(x) will be periodic if a +ve real number T exists such that
( ) ( )
fx T fx , + = ?× ?Domain.
There may be infinitely many such real number T which satisfies the above equality. Such a least +ve number 
T is called period of f(x).
(i) If a function f(x) has period T, then period of f(xn+a)=T/n and period of (x/n+a)=nT.
(ii) If the period of f(x) is T
1 
& g(x) has T
2
 then the period of f(x) ± g(x) will be L.C.M. of T
1
& T
2
 provided it satis-
fies definition of periodic function.
(iii) If period of f(x) & f(x) are same T, then the period of af(x)+bg(x) will also be T.
Function Period
sin x, cos x
2 p
sec x, cosec x
tan x, cot x
p
sin (x/3)
6 p
tan 4x
p /4
cos 2 p x
1
cosx p
sin
4
x+cos
4
x
p /2
2 cos
x
3
? ? - p
? ?
? ?
6 p
sin3 x + cos
3
x
2 p /3
Sin
3
 x +cos
4
x
2 p
sinx
sin5x
2 p
2 2
tan x cot x - p
x-[x] 1
[x] 1