Question 40
Determine all functions $f: \mathbb{R} \to \mathbb{R}^+$ satisfying the conditions
$$
i.\ f(x^2) = f(x)^2 - 2x f(x),$$
$$
ii.\ f(-x) = f(x - 1),$$
$$
iii.\ 1 \lt x \lt y \text{ implies } f(x) \lt f(y).$$
for all real numbers $x$ and $y$.
$$
i.\ f(x^2) = f(x)^2 - 2x f(x),$$
$$
ii.\ f(-x) = f(x - 1),$$
$$
iii.\ 1 \lt x \lt y \text{ implies } f(x) \lt f(y).$$
for all real numbers $x$ and $y$.