In need help understanding this (It's from Evan Chen's Introduction to Functional equations)
When trying to obtain injective or surjective, watch for "isolated" variables or parts of the equation. For example, suppose you have a condition like $$f(x+2xf(y)^2)=yf(x)+f(f(y)+1)$$. Noting that $f \equiv 0$ works assume $f$ is not $0$ everywhere. Then by taking $x_0$ with $f(x_0) \neq 0$, one obtains that $f$ is injective (try putting in $y_1$ and $y_2$)
I have tried setting $f(y_1)=f(y_2)$ and trying to plug $y_1$, $y_2$ and $x_0$ in but i have not been getting anywhere, I feel like I'm missing something pretty obvious. Could someone explain why this implies injectivity and how to approach the general case of isolated variables?
Best Answer
I'll use $x$ and $y$ such that $f(x) = f(y)$, because it's easier to type. Then $$f(x + 2x f(x)^2) = y f(x) + f(f(x) + 1)$$
As long as $f(x) \not = 0$, that is a linear equation in $y$, so it has exactly one solution. Since $f(x) = f(x)$, we know $y = x$ must be a solution, so it is the only solution.