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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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3 Answers

Best answer
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$f(x) = x – [x]$
$f(x)$ 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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  • 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:-

     
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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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