What is the f, if f(f(x)) = 2x^2 -1

[Xamfy19]

Ok, I have tried to solve this since yesterday. I tried inverse function, derivative.. still going nowhere. Please help.

 

[jcsd]

Just try working through it:

$$f\circ f(x) = x^2 \therefore f(x) = x^{\sqrt{2}}$$

$$f\circ f(x) = 2x^2 \therefore f(x) = 2^{\frac{1}{1 + \sqrt{2}}}x^{\sqrt{2}}$$

edited to add: I've left the rest for you (I think you can obtian the answer this way, but didn't work it all the way through so I can't say for sure).

edited to add again: actually there doesn't seem to be any easy way to get the answer this way.

 

[Xamfy19]

Thanks a lot, jcsd.