I want to know how to find the components of a non linear composite function. I know how to find the components of some linear composite functions but when it comes to non linear composite functions i cant wrap my head around it. For example if i have $g \circ h$ $=$ $e^{-x^3}$ then how do I find $h(x)$ and $g(x)$. Is there a method to do so? If there is are there some exceptions to which i cant find the components?
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Since $e^{-x^3}=\exp\left(-x^3\right)$, you can take $g(x)=\exp(x)$ and $h(x)=-x^3$. – José Carlos Santos Jun 22 '22 at 18:02
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I think there can be infinitely many such pairs: take for eg. 1. f= $e^{-x}, g=x^3$ 2. f= $e^{x}, g=-x^3$, 3. f= $e^{-\sqrt[3]{x}}, g=x^9$, f= $\frac1x, g=e^{x^3}$ 4. f= $e^{x^3}, g=-x$ etc. – insipidintegrator Jun 22 '22 at 18:07
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@JoséCarlosSantos thank you. – UnparalledDumbness Jun 22 '22 at 18:09
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@insipidintegrator if so , is there a more general and rigorous way of finding the components when conditions on $f$ and $g$ are given rather than random guessing? – UnparalledDumbness Jun 22 '22 at 18:11
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Think of what you'd do to a specific input: say $5$. First you would compute $-5^3$, and then put that result into the exponentiation function to get $e^{-5^3}$.
This suggests that taking $h(x)=-x^3$ and $g(x)=e^x$, which gives $g\circ h (x)=g(-x^3)=e^{-x^3}$, the desired function.
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