Question

Se considera functiile f: R ⇒ R; g: R ⇒ R definite respectiv prin formulele:
f(x)= x² + x - 1;
g(x)= y² - y + 1.
Sa se determine g compus cu f si f compus cu g.

Answer 1

(fog)(y)=f(g(y))=$g^2(y)+g(y)-1=(y^2-y+1)^2+y^2-y+1-1=y^4+y^2+1$-$2y^3+2y^2-2y+y^2-y=y^4-2y^3+4y^2-3y+1$
(gof)(x)=g(f(x))=$f^2(x)-f(x)+1=(x^2+x-1)^2-x^2-x+1+1=x^4+x^2+1+$+$2x^3-2x^2-2x-x^2-x+2=x^4+2x^3-2x^2-3x+3.$