`[x]` doesn't usually mean absolute value:

Generally, although I don't think this notation is totally standardized, you have the following:

`|~ x ~|` is the ceiling of x, and it means round up to the nearest integer, e.g:

`|~ 2.34 ~| = 3 `, `|~ -1.4 ~| = -1 `, `|~ 2 ~|=2`,

`|__ x __|` is the floor of x, and it means round down to the nearest integer, e.g:

`|__ 2.34 __| = 2`, `|__ -1.4 __| = -2`, `|__ 2 __| = 2`

`[x]` is the nearest integer to x, so it is the regular old rounding from elementary school (where you round up if you are at a .5) :

`[2.3] = 2` `[2.4999] = 2`, `[2.5]=3`

So, for some examples:

`1-[1] = 1-1 = 0` (this will happen with any integer)

`2.3-[2.3]=2.3-2=.3`

`4.8-[4.8]=4.8-5=-.2`

`3.5-[3.5]=3.5-4=-.5`

The graph looks like:

If the function is x -abs(x), then the graph will be:

When x is positive the graph coincide with the x axis that why it is not showing.

To obtain the above graph we can look at the follwoing table of values:

x=-3 => -3-abs(-3)=-3-(3)=-6

x=-2 =>-2-abs(-2)=-2-2=-4

x=-1 => -1-abs(-1)=-1-1=-2

x=0 => 0-abs(0)=0

x=1 => 1-abs(1)=1-1=0

x=2 => 2-abs(2)=2-2=0

To study the above graph in more depth we can notice the following:

Since the absolute value of positive numbers don't change the numbers we can divide the problem into two parts:

When x>0 or x=0, then x-abs(x)=x-x=0. Its graph is simply coincide with the x-axis.

When x<0, then x-abs(x)=x-(-x)=2x. Hence for all negative values we need to graph 2x.

whether x is positive or negative [x] is a positive value.

eg: [1] = 1 and [-1] = 1

f(x) = x-[x]

When x = -3 then;

f(-3) = -3-[-3] = -3-3 = -6

In the same manner we can calculate the rest.

f(-3) = -6

f(-2) = -4

f(-1) = -2

f(0) = 0

f(1) = 0

f(2) = 0

f(3) = 0

So when x<=0 we can say x=-x

f(x) = -x-[-x] = -x-x = -2x

So when x>0 we can say x=x

f(x) = x-[x] = x-x = 0

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