Forgive me in advance for this probably stupid question. I was thinking about when we restrict a given function $f(x, y)$ to a curve, in order to study certain properties in an easier way. For example when we have to optimise a function on a constraint and we have the possibility to study the restriction of $f$ on that constraint.
Anyway the question is this: say I have $f(x, y) = x^2 + y^2$ and for some reason I am going to study the restriction of $f$ along the curve $y = \sqrt{x}$. This leads to $f\big|_{y = \sqrt{x}} = f(x) = x^2 + x$.
The restriction of $f(x, y)$ over $y = \sqrt{x}$ is a parabola (convex), but here are the two main doubts:
Choosing $x = -\frac{1}{2}$ leads to $f(-1/2) = -1/4$. This is a negative value. How to interpret this result, in the light of the fact that $f(x, y)$ is instead always non negative?
Geometrically: how can I see the restriction? I tried to imagine the paraboloid and the square root, but even plotting them I wasn't able to clearly see the "new" parabola.
Sorry again for these stupid questions, I am a beginner in these topics (convex analysis et al) and I want to dig the deeper possible.