The definition of periodic is:

There is some number, p, such that f(x)=f(x+p) for all x

Visually, if you graphed the function, then picked up the graph and moved the whole thing to the left p units, the new graph would look the same as the original. Picture the sine graph. If you pick the curve up off the paper, and move it `2pi` over, and set it back down, it looks the same as it did before you moved it:

So, to your question:

`{x}` means the non-whole-number part of your graph. So, for example, {6.05} = .05, {7.9898}=.9898, {2}=0

Technically, it means `x- |__ x __|`

which is the number minus the largest whole number less than or equal to the number

which means, for negative numbers: {-2.4} = -2.4-(-3) = .6

So, this is a periodic function, with period 1:

think of an example: {4.23}=.23 , {5.23}=.23 , {6.23}=.23

f(.23) = f(.23+1) = f(1.23 + 1) = ...

The graph looks like:

Next, `"cot" (x+7)` is also periodic. This is because tan is periodic, so cot = 1/tan is periodic. The +7 moves the graph 7 units left. But what remains is still a picture that repeats every `2pi`

The graph looks like:

Thus, it is the last function that is not periodic.

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