Question

Para a função f, de R em R, definida por f(x) = 3x:
a) determine x, se f(x²) = f(2x + 3).
b) verifique se f(3x + 4) = 3f(x) + 4.

Answer 1

a)

Vamos começar determinando f(x²) e f(2x+3) separadamente para facilitar.

$f(x^2)~=~3\,.\,(x^2)\\\\\\\boxed{f(x^2)~=~3x^2}\\\\\\\\f(2x+3)~=~3\,.\,(2x+3)\\\\\\\boxed{f(2x+3)~=~6x+9}$

Igualando f(x²) e f(2x+3):

$3x^2~=~6x+9\\\\\\3x^2-6x-9~=~0\\\\\\\boxed{x^2-2x-3~=~0}\\\\\\Aplicando~Bhaskara\\\\\\\Delta~=~(-2)^2-4.1.(-3)~=~4+12~=~\boxed{16}\\\\\\x~=~\dfrac{2\pm\sqrt{16}}{2~.~1}~=~\dfrac{2\pm4}{2}~~~\left\{\begin{array}{ccc}x'&=&3\\\\x''&=&-1\end{array}\right.$

Logo "x" pode ser 3 ou -1.

b)

Vamos, como feito na (a), calcular separadamente.

$f(3x+4)~=~3\,.\,(3x+4)\\\\\\\boxed{f(3x+4)~=~9x+12}\\\\\\\\3f(x)+4~=~3\,.\,(3x)+4\\\\\\\boxed{3f(x)+4~=~9x+4}$

Sendo assim, vamos que f(3x + 4) é diferente de 3f(x) + 4.

$\boxed{9x+12~\ne~9x+4}$