Consider the graphs of $f(x)=x^2$ and $g(x) = x^2 + \frac{1}{x^2 + 1}$:
It seems to me that $g(x)$ looks like it is $f(x)$ looked at from a specific angle in 3D, where $V$ is some vector perpendicular to the $x,y$ plane:
Is there any way to prove whether this is the case, or whether, actually, it is impossible for it to be the case?
I'd be interested in any resources I could use to learn about such relationships. Thank you.
To be precise we need an explicit model of what "projection" means. One natural definition is perspective projection, which it sounds as if you have in mind: Pick an arbitrary plane $P$ and a point $O$ not on $P$, and imagine a light source at $O$ casting shadows on $P$. (In more detail, map each point $x \neq O$ to the intersection of the line $\overline{Ox}$ with $P$. We can either tolerate that this mapping is undefined where $\overline{Ox}$ is parallel to $P$, or we can work in "projective space" where "parallel lines meet at infinity" in a precise sense.)
In this model, a parabola is a conic (algebraically a curve of degree two, geometrically the intersection of a right circular cone and a plane), so it not equivalent to the graph of $g$, which is not a conic.