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I know the form of f(x) and that of g(x) and I would like to find an expression for the function H such that H[g(x)] = f(x). f is a polynomial and g is more complex and involves some exp and cos. Is there any procedure to find H?

More details:

$f(x) = Ax^6+Bx^{12}$

$g(x) = e^x\left(\cos{x}+1\right)$

I know (numerically) g(x) and f(x) and, I would like to find a function such that H[g(x)] = f(x) without having to evaluate it from x.

Thank you,

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  • $\begingroup$ There is no general procedure for this sort of thing. You will have to be much more specific and tell us what is the domain and codomain of $H$, and how $f$ is determined from $g$. $\endgroup$ – Ben W Jan 2 at 17:57
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    $\begingroup$ For example, if $f(x)=x$ you are asking for a universal expression which gives an inverse to $g(x)$. In general, though, inverse functions can be quite hard to construct. $\endgroup$ – lulu Jan 2 at 18:09

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