The function that associates each real number to it self is called the identity function and denoted by I

I(x) = x : x R

The domain and range of the identity function are both equal to R. the graph is a straight line passing through the origin and inclined at an angle of with x axis

F(x) = x

F(1) = 1

F(2) = 2

Modulus Functions:-

If f(x) = |x| = {x when x ≥ 0,

-x when x < 0}
Is called the modulus function or absolute value function the domain of the modulus function is the set of all real number. & range is the set of all non negative real number
= {x R : x ≥ 03}

(1) If x ≥ 0 the graph coincide. With the graph of

the identity functional (g= x)

If x < 0 it is coincide to the line y = -x

**Properties of Modulus Function**

1) For any real number x

= |x|

2) a,b is a positive real number, then

≤ |x| ≤ a -a ≤ x ≤ a

≥ |x| ≥ a x ≤ -a or x ≥ a

< |x| < a -a < x < a
> |x| > a x < -a or x > a

≤ ≤ a≤ |x| ≤ b x [-b, -a] u [a,b]

< < a ≤ |x| ≤ b [-b, -a] u [a,b]

3) for real number x and y

|x+y| = |x| + |y| (x≥0 and y≥ 0) or (x< 0 and y< 0)|x-y| = |x| - |y| (x≥0 and |x|≥ |y|) or (x≤ 0 , y≤ 0 and |x| ≥ |y|)|x± y| ≤ |x| + |y||x± y| ≥ |x| - |y|

**Greatest Integer Function (Floor Function)**

f(x) = [x] for all xεR or ⌊x⌋ is called greatest integer function

for any real number x the smallest [x] to denote the greatest integer less than or equal to x

Domain of the greatest integer function is the set of R to all real number and the range is the set fo Z of all integers as it’s attains only integer value.

e.g. [2.75] = 2, [3]= 3, [0.74] = 0, [-7.45] = -8 etc

Properties :- if n is integer and x is a real number between n and n+1 then

[-n] = -[n]

[x + k] = [x] + k for any integer k

[-x] = -[x] -1

[x] + [-x] =

[x] – [-x] =

[x] ≥ k = x ≥ k where k z

[x] ≤ k = x < k where k z

[x] > k = x ≥ k +1 where k z

[x] < k = x < k where k z

[x+y] = [x] + [y+x-[x] for all x,y R

= [n x], n N

**Smallest Integer Function**

f(x) = ⌈x⌉ for all x εR

Is called the smallest integer function or the ceiling function.

The domain of the smallest integer function is the set of R of all real number and its range is the set z 0 < all integers e.g :- ⌈4.75⌉=5, ⌈-7.2⌉ = -7,⌈5⌉=5, ⌈.75⌉=1 etcProperties of Smallest Integer Function ⌈-n⌉= -⌈n⌉ ,n z

⌈-x⌉= -⌈x⌉+ 1,x R-z

⌈x+n⌉= ⌈x⌉+ n,xεR-z and n z

⌈x⌉+ ⌈-x⌉ =

⌈x⌉- ⌈-x⌉ =

**FRACTIONAL PART FUNCTION**

The function f(x) = {x} for all xεR

The symbol {x} denote the fractional part of or decimal part of x.

The domain of the fractional part function is the set of R of real number and range is the set [0,1]

f(x) = {x} = x – [x] : x εR

e.g. 3.45 = 0.45, [2.75] = 0.25, [-0.55] = 0.45, [3] = 0, [-7] = 0 etc

**SIGNUM FUNCTION**

If f(x) = .3 × .3 =.09

f(x) =

The domain of the signum function is the set of R (all real number) and the domain is the set

(-1,0,1)

EXPONENTIAL FUNCTION:-

F (x) = where a> 0 and a ≠ 1

If a > 1

If y = f(x) =

We observe that –

If 0 < a<1Y = f(x) = decrease with the increase in x.

And y>0 : x ε R

if then graph