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Piecewise function is considered one the most difficult concept.
18 Questions around this concept.
If and the equation
(where
denotes the greatest integer
) has no integral solution, then all possible values of a lie in the interval :
Signum function:
The function f : RR defined by
is called the signum function. The domain of the signum function is R and the range is the set {-1,0,1}.
This function can also be written in another form:
Graph:
Greatest integer function (G.I.F.)
The function f: R R defined by f(x) = [x], x
R assumes the value of the greatest integer which is equal to or less than x. Such a functions is called the greatest integer function.
eg;
[1.75] = 1
[2.34] = 2
[-0.9] = -1
[-4.8] = -5
[4] = 4
[-1] = -1
Graph:
From the definition of [x], we
can see that
[x] = –1 for –1 x < 0
[x] = 0 for 0 x < 1
[x] = 1 for 1 x < 2
[x] = 2 for 2 x < 3 and so on.
Properties of greatest integer function:
i) [ a ] = a (If a is an integer)
ii) [[x]] = [x]
iii) x-1 < [x] ≤ x
iv) [ x + a ] = [ x ] + a (If a is an integer)
v) [ x - a ] = [ x ] - a (If a is an integer)
vi)
Fractional part function:
When [ x ] is the Greatest Integer Function
Eg
{2.2} = 2.2 - [2.2] = 2.2 - 2 = 0.2
{1.7} = 1.7 - [1.7] = 1.7 - 1 = 0.7
{2} = 2 - [2] = 2 - 2 = 0
{ - 2.2} = -2.2 - [-2.2] = 2.2 - (-3) = 0.8
{ - 1.7} = -1.7 - [ - 1.7] = 1.7 - (-2) = 0.3
{ - 2} = - 2 - [-2] = 2 - ( - 2) = 0
Clearly , 0 ≤ {x} < 1
Graph
Domain: R
Properties of fractional part of x
i) {x} = x if 0 ≤ x < 1
ii) {a} = 0, if a is an integer
iii) 0 ≤ {x} < 1
iv) { x + a } = {x} ( If a is an integer)
v) {x} + {-x} = 1, if x doesn’t belongs to integer
vi) {x} + {-x} = 0, if x belongs to integer
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