Question

f(x)=ax+b
Aratati ca f(x/2)+f(y/2)=f(x+y/2)+b

Answer 1

$f(x) = ax+b\\ \\ f\Big(\dfrac{x}{2}\Big) +f\Big(\dfrac{y}{2}\Big) = a\cdot \Big(\dfrac{x}{2}\Big)+b+ a\cdot\Big(\dfrac{y}{2}\Big)+b = \\ \\ = a\cdot \Big(\dfrac{x}{2}+\dfrac{y}{2}\Big)+b+b = a\cdot \Big(\dfrac{x+y}{2}\Big)+b+b = \\ \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{ a\cdot \Big(\dfrac{x+y}{2}\Big)+b = f\Big(\dfrac{x+y}{2}\Big)} \\ \\ =f\Big(\dfrac{x+y}{2}\Big)+b$

$\Rightarrow f\Big(\dfrac{x}{2}\Big) +f\Big(\dfrac{y}{2}\Big) = f\Big(\dfrac{x+y}{2}\Big)+b$