[seanhunter]

When you say they are not even functions, are you saying because they are not everywhere bijective? Because they look like rotated parabolas to me which means they will have continuous first and second derivatives which is (I think) all you need for Newton’s method isn’t it?

[casualrandomcom]

You are thinking of those parables as parametric curves, I guess, but Newton's method is about approximating an arbitrary function by its Taylor expansion truncated to the second degree, which is a polynomial function of second degree. The graphs of these functions can be thought as parametric curves, but (my point is also) these are not those that were drawn in the illustration.

[MITSardine]

I think the person meant that, to a single x, they map more than one value. A function needs to have at most one image to the antecedent. In other words, it can't have "vertical stretches". This is different from bijectivity (specifically, injectivity here) which asks that two different x values never map to the same y.