# Find the form of functional

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,

• 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 $g$. – Ben W Jan 2 at 17:57
• 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. – lulu Jan 2 at 18:09