f(x) can be either an even function or an odd function.
If f(x) was an odd function, then f(x) + f(-x) = 0, but here it has been given that it has degree 10. So, it must be an even function.
Therefore, f(x) = f(-x) => f'(x) = -f'(-x)
Also, it has been mentioned that g(x) is the derivative of f(x).
So, g(x) = f'(x) and g(-x) = -f'(-x) => g(x) - g(-x) = f'(x) - (-f'(-x)) => g(x) - g(-x) = f'(x) + f'(x) => g(x) - g(-x) = 2 * f'(x) But, f'(x) will have degree 9.