Problem: Suppose that $f:\;\mathbb{R}^2\to\mathbb{R}^2$ is an injective mapping from the 2-dimensional Euclidean plane into itself which maps lines into (instead of onto) lines and whose range contains three non-collinear points. Can we say that f is an affine transformation?

Note that neither of two "into"s in assumptions means "onto". If either of two "into"s replaced by "onto", then the anwer is yes which can be deduced from the result in Li Baokui and Wang Yuefei's paper, and Chubarev and Pinelis' paper respectively.

On the other hand, if removing the injectivity, we can construct certain counterexample.

Counterexample: $f:\mathbb{R}^2→\mathbb{R}^2.$ Let f fix every point in some line $L$ and map the complement to one point outside $L$. $f$ is not an affine transformation.