SIgnum , Modulus and Greatest Integer Functions

7. Signum function


, if


, if


, if


Domain =





Signum function takes the sign of the x


e.g.

,


Clearly the Signum Function is defined for all Real values i.e. all values of 'x' but it can attain only three values -1,0 and 1.



8. Modulus Function


, if


, if


Domain = R










Properties of Modulus:For revision, here is the quick look of the properties of modulus function


1.

2.

3.

4.



9. Greatest integer function f(x) = [x]


It is defined as the largest integer less than or equal to 'x'


For example,

, [5] = 5 , [ -8] = -8 etc.


consider 1.3


Step 1 : which integers are less than or equal to 1.3


clearly, .................... -3, -2, -1, 0 and 1


Step 2 : Among the above integers which is greatest


clearly it is 1


so, [1.3] = 1


Now, consider -1.3


Step 1: which integers are less than or equal to -1.3


............ -4, -3 and -2


Step 2 : Among the above integers which is greatest


clearly -2 is the greatest


so, [-1.3] = -2


check yourself,


a) [28.5] = ?


b) [-0.0009] = ?


c) [0.99] = ?


d) [-11.78] = ?


e) [-100003] = ?


To draw its graph, we note that


f(x) = -1, if


f(x) = 0 , if



      = 1 , if


............ and so on


Domain = R


Range = I


Properties of [x] :


1. , I is integer






2.






3.






1. consider, , where I belongs to integers,


let and


L.H.S. =


R.H.S. =

so, if there is any question like then






2. Consider,


Since x is not integer, let





and,






3. Consider,


Since x is integer, let




















Lets see the graph again,









It is clearly seen from the graph that function y = [x] is embedded between two lines y = x and y = x - 1,


if x = I , [x] = x, i.e. function y = [x] and y = x become equal at integers, that is why function touches the y = x line.

and if , , check let x = 3.5 => [x] = 3 and x - 1 = 3.5 - 1 = 2.5,

therefore, , if

and , if


MOST IMPORTANT PROPERTIES


1. , [x] never equals to x - 1 but [x] equal to x when x is integer.


2. ,, I is integer


3. , I is integer


But,