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1. 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!
2. (Original post by ComputerMaths97)
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!
You might find it easier to think through this if you remember the identity

This of necessity takes you into the complex plane (ln(-1) = i Pi) but will allow you to investigate things systematically.

In fact, if , then and therefore

which is probably a lot easier to think about!

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