Maths Uni people needed!

So I've been thinking a lot lately about the graph of y = (-1)^x. I'm in year 13 so as you'd assume, I can't do much other than cover special cases.

If x is integer: If x is odd, y = -1
If x is even, y = 1
If x is rational:
Let n represent any number.
If x can be expressed as n / 2n + 1 then:
If n is even, y = 1
If n is odd, y = -1

So basically, the only cases that work in the real planes are when x can be expressed as a rational number with an odd denominator... Is this it?

I asked my Further Maths teacher and he said go research the gamma function, but that didn't seem to answer any questions. Do I need to research further?

Any info on this graph would be helpful, thank you!