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,FUNCTIONS
, MARKS 3
FU N CT I ON S
El e m e n t a r y N u m b e r Sy s t e m
The whole of calculus is based on the concepts of real numbers. So let us briefly discuss real
numbers.
Real Numbers
Rational Numbers Irrational Numbers
Integers Fractions
Positive O Negative
Integers Integers
Term s & t heir definitions
Ć Integers : The numbers ă 4, ă 3, ă 2, ă 1, 0, 1, 2, 3, 4 are called integers.
i.e. set of positive integers + zero + negative integers.
They are denoted by I or Z
Ć Natural numbers : They are a subset of integers & denoted by N
N = {1, 2, 3, .... }
i.e. all positive integers.
Ć Whole numbers : It is also a subset of integers. It contains positive integers + zero.
Denoted by W
W = {0, 1, 2, 3 ...}
Ć Zero : Zero is an integer but neither a positive nor a negative integer. But it
is non-negative as well as non-positive integer.
I nt e r va l s
1. Open Interval : For two real numbers a and b, where a < b, the set of all real numbers
lying strictly between a and b (i.e. not including a and b) is called an
open interval.
denoted by ( ) [round brackets]
i.e. a < x < b x (a, b)
FUN C TIO N S
, 4 MARKS
2. Closed interval : Again for same 2 real numbers, if x can take values between a and
b, including a & b, then its a closed interval.
i.e. a x b x [a, b] {square brackets are used}
3. Half Open Interval : It contains both type of intervals, open, closed interval & closed open
interval. In this type only one end point is included.
a < x b x (a, b]
a x < b x [a, b)
4. Infinite intervals : Before going to intervals let us discuss first about infinity, denoted
by .
By infinity we mean that it is a very big real number, larger than any
real number but how large, it is not fixed.
When we say x R, we indirectly mean
ă < x < or x (ă , )
coming to infinite intervals now,
whenever is at one or both the end points we never include them;
i.e.
ă < x < or x (ă , )
round brackets
Not square brackets
ă < x < a or x (ă , a)
ă < x a or x (ă , a]
x a or x (ă , a]
So m e B a s i c De f i n i t i o n s
Ć Domain : For a given function y = f (x), the set of values which x can take provided that
for those values y is well defined, is known as Domain of the function.
1
for ex. y , here x can take all real values except 0 because at x = 0 the value
x
of y is invalid.
Ć Range : For a given function y = f (x), the set of values which y can take, corresponding
to each real number in the domain, is known as Range of function.
for ex. y = x2 , here x can take all real values but y can take only positive values.
Domain & Range can also be expressed as
Domain Range
function
The values which The values which
input can take output can take
FUN C TIO N S
JEE Main JEE Adv. BITSAT WBJEE MHT CET and more...
4.8 50,000+ 2,00,000+
Rating on Google Play Students using daily Questions available
With MARKS app you can do all these things for free
Solve Chapter-wise PYQ of JEE Main, JEE Advanced, NEET, BITSAT, WBJEE, MHT CET & more
Create Unlimited Custom Tests for any exam
Attempt Top Questions for JEE Main which can boost your rank
Track your exam preparation with Preparation Trackers
Complete daily goals, rank up on the leaderboard & compete with other aspirants
4.8 50,000+ 2,00,000+
Rating on Google Play Students using daily Questions available
,FUNCTIONS
, MARKS 3
FU N CT I ON S
El e m e n t a r y N u m b e r Sy s t e m
The whole of calculus is based on the concepts of real numbers. So let us briefly discuss real
numbers.
Real Numbers
Rational Numbers Irrational Numbers
Integers Fractions
Positive O Negative
Integers Integers
Term s & t heir definitions
Ć Integers : The numbers ă 4, ă 3, ă 2, ă 1, 0, 1, 2, 3, 4 are called integers.
i.e. set of positive integers + zero + negative integers.
They are denoted by I or Z
Ć Natural numbers : They are a subset of integers & denoted by N
N = {1, 2, 3, .... }
i.e. all positive integers.
Ć Whole numbers : It is also a subset of integers. It contains positive integers + zero.
Denoted by W
W = {0, 1, 2, 3 ...}
Ć Zero : Zero is an integer but neither a positive nor a negative integer. But it
is non-negative as well as non-positive integer.
I nt e r va l s
1. Open Interval : For two real numbers a and b, where a < b, the set of all real numbers
lying strictly between a and b (i.e. not including a and b) is called an
open interval.
denoted by ( ) [round brackets]
i.e. a < x < b x (a, b)
FUN C TIO N S
, 4 MARKS
2. Closed interval : Again for same 2 real numbers, if x can take values between a and
b, including a & b, then its a closed interval.
i.e. a x b x [a, b] {square brackets are used}
3. Half Open Interval : It contains both type of intervals, open, closed interval & closed open
interval. In this type only one end point is included.
a < x b x (a, b]
a x < b x [a, b)
4. Infinite intervals : Before going to intervals let us discuss first about infinity, denoted
by .
By infinity we mean that it is a very big real number, larger than any
real number but how large, it is not fixed.
When we say x R, we indirectly mean
ă < x < or x (ă , )
coming to infinite intervals now,
whenever is at one or both the end points we never include them;
i.e.
ă < x < or x (ă , )
round brackets
Not square brackets
ă < x < a or x (ă , a)
ă < x a or x (ă , a]
x a or x (ă , a]
So m e B a s i c De f i n i t i o n s
Ć Domain : For a given function y = f (x), the set of values which x can take provided that
for those values y is well defined, is known as Domain of the function.
1
for ex. y , here x can take all real values except 0 because at x = 0 the value
x
of y is invalid.
Ć Range : For a given function y = f (x), the set of values which y can take, corresponding
to each real number in the domain, is known as Range of function.
for ex. y = x2 , here x can take all real values but y can take only positive values.
Domain & Range can also be expressed as
Domain Range
function
The values which The values which
input can take output can take
FUN C TIO N S
