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Calculus Level 5

If $$f(x)$$ and $$g(x)$$ are functions defined in the real domain and co-domain, such that $$\sqrt{1-(f(x))^2}=g(x)$$, which of the following statements are necessarily true?

• A: If $$g(x)$$ is periodic with period 1, $$f(x)$$ is periodic with period $$\dfrac{1}{2}$$
• B: If $$g(x)$$ is a continuous function, $$f(x)$$ is also continuous in their respective domains.
• C: If $$-f(c)=f'(c)=0.5;\dfrac{g'(c)}{g(c)}=\dfrac{1}{3}$$.
• D: If $$g(x)$$ is an even function,$$f(x)$$ is odd.
• E: If $$g(x)$$ is an odd function, $$f(1024)$$ is of unit modulus.

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