Question

Dado que, para todo x real, f(x) + x. f(2 - x) = x2 + 2, calcule:
a) f(1)
b) f(2)​

Answer 1

$\text{f(x) + x . f}(2-\text x) =\text x^2+2$

item a)

f(1) = ?

fazendo x = 1 :

$\text f(1) + 1.\text f(2-1) = 1^2 +2 \\\\ \text f(1) +\text f(1) = 1+2 \\\\ 2. \text f(1) = 3 \\\\ \displaystyle \huge\boxed{\text f(1) = \frac{3}{2}}\checkmark$

item b)

f(2) = ?

$\text{f(x) + x . f}(2-\text x) =\text x^2+2$

fazendo x = 0

$\text f(0) + 0.\text f(2-0)=0^2+2 \\\\ \text f(0) + 0 = 2 \\\\ \underline{\text f(0) = 2}$

fazendo x = 2 :

$\text f(2) + 2.\text f(2-2) = \text 2^2+2 \\\\ \text f(2) + 2.\text f(0) = 4+2 \\\\ \text f(2) + 2.2 = 6 \\\\ \text f(2) = 6-4 \\\\ \huge\boxed{\text f(2) = 2}\checkmark$