Find pdf of $f(x)$ such that $g(x)/f(x)$ is approximately a constant

Question

My friend asked me a question that asks to find a pdf function $f(x)$ such that $f(x)/g(x)$ is approximately a constant, where $g(x)=\sqrt{e^{x^2}+e^x}$, and $f(x) \neq g(x)$. And the range of x is [-1,2] for both f(x) and g(x).

I have no idea what should I do this, can someone give me some hints (not need to be full solution).

Thanks.

Answer 1

You could write $$ g(x) = \left( \sum_{n=0}^\infty \frac{x^{2n} + x^n}{n!}\right)^{1/2} $$ and then just choose $f$ to be $$ f(x) \propto \left( \sum_{n=0}^N \frac{x^{2n} + x^n}{n!}\right)^{1/2} $$ for an $N$ that gives you the desired level of closeness.