Set Identity function

The set Identity function and related functions are discussed below.

What is Set Identity function?

The function that associates each real number to itself is called the identity function and is denoted by the I
$I : R \rightarrow R$
I(x) = x : x $\varepsilon$ R

Domain and range of the Set Identity function

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 an 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 numbers & range is the set of all nonnegative 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 coincides with 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
The domain of the greatest integer function is the set of R to all real numbers and the range is the set of Z of all integers as it attains only integer value.

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

Properties:- if n is an 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 numbers and its range is the set z 0 < all integers

e.g:- ⌈4.75⌉=5, ⌈-7.2⌉ = -7,⌈5⌉=5, ⌈.75⌉=1, etc.

Properties 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} denotes the fractional part of or decimal part of x.
The domain of the fractional part function is the set of R of real numbers and the 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 numbers) 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<1 Y = f(x) = $a^x$ decrease with the increase in x. And y>0 : x ε R
$a^x$ $\left\{\begin{array} > 1 : x < 0 \\ =1 : x= 0 \\ <1 for x > 0 \end{array}\right.$