General formula of a composite quadratic function

Question

I've already found a general formula to composite linear functions, i.e., if I have a function $f(x)=ax+b$, I can find a formula to $f^{(n)}(x)$ as function of the constants $a,b$ and $n$.

I want to know if it's possible to find a formula to the n-composite function of $g(x)=ax^2+bx+c$.

score 2 · Accepted Answer · answered Apr 08 '18 at 23:37

2

It is possible but computations get very cumbersome. I recommend starting with derivatives and then integrate the results.

for example let $$f(x)=g(g(x))$$

Then $$f'(x)=g'(g(x))g'(x)= (2ag(x) +b)(2ax+b) = 4a^2xg(x)+2ab (x+g(x))+b^2$$

Simplify and integrate and find the constant of integration by $x=0$.

answered Apr 08 '18 at 23:37

Mohammad Riazi-Kermani

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I didn't understand, when I said $f^{(n)}(x)$ I meant the n-composite instead of n-derivative. Thank you for your answer – user42912 Apr 09 '18 at 00:42
I know. I told you that it is easier if you first take derivative of the nth composite and integrate the result. The derivative of the iterations is just the product of derivatives which in this case is very simple. Look at the example please. – Mohammad Riazi-Kermani Apr 09 '18 at 00:57
I got it. Thank you very much! – user42912 Apr 09 '18 at 01:20
Thanks for a very good question. – Mohammad Riazi-Kermani Apr 09 '18 at 01:33

General formula of a composite quadratic function

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