Question about affine transformation and one-one function. An Affine transformation is a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(v) = Av+b $, where $\det A \neq 0 $ and $b \in \mathbb{R}^2.$ 
My professor give me a task, I need to prove that a function $f$ is Affine transformation iff $f$ is one-one.
It's a little confuse, I have proved that if $f$ is affine then $f$ is one - one, but the other hand I don't know and I suspect it's not true. Of course, if I take $f: \mathbb{R}_{+} \rightarrow \mathbb{R},$ where $f(x) = x^2,$ then this function is one- one and not affine, but I need some example in $\mathbb{R}^2$ with the form $f(v)=Av + b.$

$f$ can be one-one but not affine.
For example, $f(x,y) = (x^3,y)$  This function is one-one, but not affine.
