Question Number 216910 by Marzuk last updated on 24/Feb/25
![Ed:.06 a function δ(x) is a composite function which is as follow [{(f ○ g)(x)} ○ {(g ○ f)(x)}] ○ [{(f ′ ○ g ′)(x)} ○ {(g ′ ○ f ′)(x)}] where f(x) = Π_(n = 1) ^∞ ((nx^3 − nx^2 − nx −n)/(n^3 x − n^2 x − nx −x)) g(x)= f ′′(x) ∫_( ψ) ^( δ) δ(x) dx ∈ R\Q ? true or false?](Q216910.png)
$${Ed}:.\mathrm{06} \\ $$$${a}\:{function}\:\delta\left({x}\right)\:{is}\:{a}\:{composite}\:{function} \\ $$$${which}\:{is}\:{as}\:{follow}\: \\ $$$$\left[\left\{\left({f}\:\circ\:{g}\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:\circ\:{f}\right)\left({x}\right)\right\}\right]\:\circ\:\left[\left\{\left({f}\:'\:\circ\:{g}\:'\right)\left({x}\right)\right\}\:\circ\:\left\{\left({g}\:'\:\circ\:{f}\:'\right)\left({x}\right)\right\}\right] \\ $$$${where} \\ $$$${f}\left({x}\right)\:=\:\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{{nx}^{\mathrm{3}} \:−\:{nx}^{\mathrm{2}} \:−\:{nx}\:\:−{n}}{{n}^{\mathrm{3}} {x}\:−\:{n}^{\mathrm{2}} {x}\:−\:{nx}\:−{x}} \\ $$$${g}\left({x}\right)=\:{f}\:''\left({x}\right) \\ $$$$\underset{\:\psi} {\int}\overset{\:\delta} {\:}\delta\left({x}\right)\:{dx}\:\in\:\:\mathbb{R}\backslash\mathbb{Q}\:? \\ $$$${true}\:{or}\:{false}? \\ $$