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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

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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