Question

se da functia f:R->R ,f (x)=-5x+3.sa se determine functia g:R->R care sarisface relatia:(f°g)(x)=10x patrat+5x-2,oricare ar fi x €R

Answer 1

$f(x) = -5x+3 \\ \\ (f\circ f)(x) = 10x^2+5x-2 \\ \\ g(x) = ax^2 + bx+c\\ \\ (f\circ g)(x) =10x^2+5x-2 \\ \\ -5(ax^2 + bx+c)+3 = 10x^2+5x-2 \\ \\ -5ax^2 -5bx-5c+3 = 10x^2+5x-2 \\ \\ \Rightarrow -5a = 10; -5b = 5; -5c+3 = -2 \Rightarrow \\ \\ \Rightarrow a = -2; b = -1; c = 1 \Rightarrow \\ \\ \Rightarrow \boxed{g(x) = -2x^2 -x +1}$