Mathematics

The first condition tells you that $f(x)$ behaves like $ax^2$ near zero—this is your anchor for understanding $f$'s growth rate. The second condition involves $f(1-\cos x)$, which means you need to substitute the argument $1-\cos x$ into that same quadratic behavior, then use the fact that $1-\cos x \approx \frac{x^2}{2}$ to extract information about $g(x)$. Once you know how $g$ behaves, scale the final expression using $1-\cos 2x \approx 2x^2$ and simplify.

The first condition tells you that $f(x)$ behaves like $ax^2$ near zero—this is your anchor for understanding $f$'s growth rate. The second condition involves $f(1-\cos x)$, which means you need to substitute the argument $1-\cos x$ into that same quadratic behavior, then use the fact that $1-\cos x \approx \frac{x^2}{2}$ to extract information about $g(x)$. Once you know how $g$ behaves, scale the final expression using $1-\cos 2x \approx 2x^2$ and simplify.