Question

Fie functiile f,g
care verifica relatille;
2f(x+2)-g(x-2)=3x-3 si
4f(x+2)+3g(x-2)=x+19
determinati functiile f si g

Answer 1

[tex] \left \{ {{2f(x+2)-g(x-2)=3x-3} \atop {4f(x+2)+3g(x-2)=x+19}} \right. \\ Pt. x\rightarrow x+4:\\ \left \{ {{2f(x+6)-g(x)=3(x+4)-3} \atop {4f(x+6)+3g(x)=x+4+19}} \right. \Leftrightarrow \left \{ {{2f(x+6)-g(x)=3x+9|\cdot 2} \atop {4f(x+6)+3g(x)=x+23}} \right. \Leftrightarrow\\ \Leftrightarrow \left \{ {{4f(x+6)-2g(x)=6x+18} \atop {4f(x+6)+3g(x)=x+23}} \right. \\ Daca\ scadem:-5g(x)=5x-5\Rightarrow \boxed{g(x)=-x+1}\\ 2f(x+6)+x-1=3x+9\\ 2f(x+6)=2x+10\\ f(x+6)=x+5\\ f(x)=x-6+5\Rightarrow\boxed{f(x)=x-1}[/tex]