y=x2 is not an injection because it is not 1-to-1: it fails the horizontal line test.
Another way of looking at it:
If our f(x) is x2 then in order for it to be injective f(x1)=f(x2) must imply x1= x2
However, f(-1)=f(1) because (-1)^2=(1)^2, but -1≠1. Therefore, it cannot be injective.
Simplest way to tell if a function is injective or not? If you know the graph of the function and every horizontal straight line passes through it once and only once, then you know it's injective.
Hope this helps! If you have any questions in any of the steps, please comment. :)