Sets Identity function

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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

$x^2$ ≤ $a^2$ |x| ≤ a -a ≤ x ≤ a
$x^2$ ≥ $a^2$ |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

$a^2$≤ $a^2$ ≤ $b^2$ 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 $\varepsilon$ NSmallest 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 $\varepsilon$R-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 $z1$
$f(x) = e^x \longrightarrow f(x) = a^x : 0< a<1$