Question

Sejam f(x)=x-3 e g(x)=x^2-2x+4. Para qual valor de x tem f(g(x))=g(f(x))?

Answer 1

$f(x)=x-3\qquad\qquad\qquad\qquad\qquad g(x)=x^2-2x+4\\\\\\f(g(x))=g(f(x))\\\\f(x^2-2x+4)=g(x-3)\\\\(x^2-2x+4)-3=(x-3)^2-2(x-3)+4\\\\x^2-2x+1=x^2-6x+9-2x+6+4\\\\x^2-2x+1=x^2-8x+19\\\\x^2-x^2-2x+8x+1-19=0\\\\6x-18=0\\\\6x=18\\\\x=\dfrac{18}{6}\\\\\boxed{x=3}\\\\\\\boxed{S=\left\{3\right\}}$