I have a function $\ f(x) = \frac12 (x^{2.657} - 0.0856x^2 + 0.376) $
The function is not invertible. It's roots are $ x=0 $ and $ x \simeq 0.22$
I would like to find a new function $ g(x) $ that has a similar shape (I am more interested to have similar values for small $x$, tough ideally It should also be $ g(0) = 0 $) but that can be inverted, perhaps like an exponential. How do I do that?
EDIT: the roots are $ x \simeq 0.619 \pm 1.17 i $
Hint
The lazy way could be to generate a table for some values of $x$ (say between $0.01$ and $0.22$ by steps of $0.01$) and perform a curve fit using$$f(x)=a+b x^c$$ Since we know $f(0)=0.188$, this gives a very good estimate of $a$.
Now, consider $$y=f(x)-0.188=b x^c\implies \log(y)=\log(b)+c\log(x)=d+c\log(x)$$ So, a linear regression gives $d,c$.
You can stop at this point or, if you want a better fit, use nonlinear regression to fine tune $a,b,c$.
It works quite well (even if its does not show the minimum value of $f(x)$ which occurs around $x=0.015$)