Showing injectivity of non-linear functions?

Question

What are some general practices for showing non-linear functions injective?

Particulary I've learned to do it with linear functions (even multiple variable), but since one cannot solve non-linear systems of equations by hand, nor do results regarding the Jacobian determinant apply, then what to do with non-linear functions?

Answer 1

Simply use the definition!

Let $f: A \rightarrow B$ be a function. Then, we call f injective if and only if:

$f(x) = f(y) \Rightarrow x = y \quad \forall x,y \in A$

or equivalent:

$x \neq y \Rightarrow f(x) \neq f(y) \quad \forall x,y \in A$

(This follows from contraposition)