Asymptotic behaviour and curve guessing

Question

So I have these data points which I plotted and found that for small $x$, my data behaves like $y=x^4$ but for large $x$, it behaves like $y=x^3$. From this, can I guess the relationship between $y$ and $x$? I tried $y=e^{-x}x^4+x^3$, but this doesn't work since the exponential doesn't kill of the $x^4$ quite enough at large values.

Answer 1

I've always liked the very simple

$$y=\frac{1}{\frac{1}{x^a}+\frac{1}{x^{b}}}$$

When $x$ is small, the $x^a$-term dominates and you get a slope of $a$ on the log-log plot, whereas when $x$ is large the $x^{b}$-term dominates with a slope of $b$ on the log-log plot.

Of course, without access to your data it's impossible to tell if the $x$-values are sufficiently large to make this feasible. However, that might be fixed with small modifications to the equation.