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10 votes
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Let $f(x) = x – [x],$ where $x\geq 0$ and $[x]$ is the greatest integer not larger than $x.$ Then $f(x)$ is a

  1.  monotonically increasing function
  2.  monotonically decreasing function
  3. linearly increasing function between two integers
  4. linearly decreasing function between two integers
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x- [x]  is also said to be the fractional part of x denoted by {x}.

 

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3 Answers

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Best answer

$f(x) = x – [x]$
$f(x)$ gives nothing but the fractional part of $x.$
Now, as we move up the number line between two consecutive integers say $a$ and $a+1$.
The value of $f(a)$ starts from $0$ and grows linearly till the consecutive integer and just before $x=a+1$(left neighbourhood of $a+1$) it tends to the value $1.$

Between $a$ and $a+1,$ the graph grows linearly from $0$ to $1$.
But, when at $x=a+1,$ the value comes back to $0$ and same linear graph continues between the next two consecutive integers and so on.
So, Answer C) linearly increasing function between two integers.

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@Puja Mishra

@abhilashpanicker29

What is meaning of monotonic increasing and How it is differed from linear increasing? 

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monotonic increasing:

function value will always increase or remain same but never decrease.

E.g. 

1,1,2,3,4,4,5,6,6,7……………

In the above the value are either increasing or remaining same  but never decreasing.

coming to  the question:

The value is first increasing from 0 to 1 and again coming back to 0 and then again increasing.

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7 votes
7 votes
  • The Greatest Integer Function is also known as the Floor Function.
  • It is written as f(x)=⌊x⌋
  • The value of ⌊x⌋ is the largest integer that is less than or equal to x

The function f(x)=x−[x]  the graph of the function looks like this:-

     
1 vote
1 vote
x =  1  1.1   1.2     1.3    1.4     1.5    ………………. 1.9  2.0

fx = 0   0,1   0.2    0.3     0.4     0.5  …………………0.9     0

 

function fx increasing linearly bw 0 and 1
Answer:

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