Piecewise function - Practice Questions & MCQ

Signum function: 

The function f : RR is defined by

$\operatorname{sgn}(\mathrm{x})=\left\{\begin{array}{ccc}1 & \text { if } & x>0 \\ -1 & \text { if } & x<0 \\ 0 & \text { if } & x=0\end{array}\right.$

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:

$\operatorname{sgn}(x)=\left\{\begin{array}{c}\frac{|x|}{x}, x \neq 0 \\ 0, x=0\end{array}\right\}$

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 function 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) $[x]+[-x]=\left\{\begin{array}{ccc}0, & \text { if } & x \in Z \\ -1, & \text { if } x \notin Z & \end{array}\right.$

 

 

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 the 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