Question

Se considera functia f:R->R, f(x)= max(2x+1, x-1). Calculati f(√2 - 2)+f(-√2 - 1).

Answer 1

[tex]f:\mathbb_{R}\rightarrow \mathbb_{R}, $ $ \quad f(x) = \max(2x+1,x-1) \\ \\ \\ f(\sqrt2 -2) +f(-\sqrt2-1) = \\ \\ =\max\Big(2\cdot(\sqrt2-2) +1,\sqrt2-2-1\Big)+\\ \max\Big(2\cdot (-\sqrt2-1)+1,-\sqrt2-1-1\Big) = \\ \\ [/tex]

$= \max(2\sqrt2-3,\sqrt2-3)+\max(-2\sqrt2-1,-\sqrt2-2) \overset{(*)}{=} \\ \\ -2\sqrt2-1-(-\sqrt2-2) = -2\sqrt2+\sqrt2-1+2 = -\sqrt2+1\ \textless \ 0 \\ \Rightarrow -2\sqrt2-1 \ \textless \ -\sqrt2-2 \\ \\ \overset{(*)}{=} 2\sqrt2-3+(-\sqrt2-2) = 2\sqrt2-\sqrt2-3-2 = \sqrt2-5 \\ \\ \\ \Rightarrow \boxed{\boxed{f(\sqrt2 -2) +f(-\sqrt2-1) = \sqrt2-5}}$