How to find a periodic or non periodic function? In the question below, which of the following is a non-periodic function?     (a) f(x)= {x} (b) f(x)= cot(x+7) (c) f(x)= 1-(sin squared x)/1 + cot x - (cos squared x)/ 1+tanx (d) f(x)= x + sin x PLS HELP !

Expert Answers

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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:

Next, I suspect there is a typo in the third expression. As it is written you have:
`f(x)=1-("sin"^2 x)/(1) + "cot" x - ("cos"^2x)/(1)+"tan" x`

It would be unusual to specify the division by 1, so I suspect there are some parentheses missing.

But whether or not the expression is exactly as written, we will use the following:

If you add/subtract/multiply/divide a bunch of periodic functions that all have the same period, what you end up with will still be periodic.

This expression (even if you add some extra parentheses) is build out of periodic functions, and so is periodic.

Finally, the last function is not periodic. sin x is periodic, x is not.

The graph looks like:  



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

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