Sets Identity function

The function that associates each real number to it self is called the identity function and denoted by I
I : R \rightarrow R
I(x) = x : x \varepsilon 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 45^0 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 R^+ = {x \varepsilon 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
\sqrt{(x^2 )} = |x|

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

x^2a^2 |x| ≤ a -a ≤ x ≤ a
x^2a^2 |x| ≥ a x ≤ -a or x ≥ a
x^2 < a^2 |x| < a -a < x < a x^2 > a^2 |x| > a x < -a or x > a

a^2a^2b^2 a≤ |x| ≤ b x \varepsilon [-b, -a] u [a,b]
a^2 < a^2 < b^2 a ≤ |x| ≤ b \varepsilon [-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] = \left\{\begin{array}-1, if x z\\0, x \varepsilon z \end{array}\right.
[x] – [-x] = \left\{\begin{array} 2[x]+1, if x z\\0, 2[x] \varepsilon z \end{array}\right.
[x] ≥ k = x ≥ k where k \varepsilon z
[x] ≤ k = x < k where k \varepsilon z
[x] > k = x ≥ k +1 where k \varepsilon z
[x] < k = x < k where k \varepsilon z
[x+y] = [x] + [y+x-[x] for all x,y \varepsilon R
[x] + [(x+ \frac{1}{n})] +  [(x+ \frac{2}{n})] + ………………..+ [(x+\frac{n-1}{n}] = [n x], n \varepsilon 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 \varepsilon z

⌈-x⌉= -⌈x⌉+ 1,x \varepsilonR-z

⌈x+n⌉= ⌈x⌉+ n,xεR-z and n \varepsilon z
⌈x⌉+ ⌈-x⌉ = \left\{\begin{array}-1, if x z\\0, x \varepsilon z \end{array}\right.
⌈x⌉- ⌈-x⌉ = \left\{\begin{array} 2[x]+1, if x z\\0, 2[x] \varepsilon z \end{array}\right.

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) = \left\{\begin{array}-\frac{|x|}{x}, x \neq 0  \\0, x= 0  \end{array}\right. .3 × .3 =.09
f(x) = \left\{\begin{array} 1, x > 0  \\0, x= 0 \\ -1 x < 0  \end{array}\right.
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) = a^x where a> 0 and a ≠ 1
If a > 1
If y = f(x) = a^x \left\{\begin{array} < 1 for x < 0  \\ =1 for x= 0 \\ >1  for x > 0  \end{array}\right.

We observe that – 2^x<3^x<4^x ……………….. : x>1
2^x = 3^x = 4^x …………………. : x = 0
2^x>3^x>4^x ……………….. : x<1

If 0 < a<1Y = f(x) = a^x decrease with the increase in x.
And y>0 : x ε R
a^x \left\{\begin{array} > 1 : x < 0  \\ =1 : x= 0 \\ <1  for x > 0  \end{array}\right.

if z<e<3 then graph

f(x) = e^x    \longrightarrow      f(x) = a^x   : a >1
f(x) = e^x  \longrightarrow        f(x) = a^x   : 0< a<1