Is there a way to rigorously define the geometrical flattening of a curve?

Question

Compare the curves $f(x) = x^2$ and $g(x) = x^2 + \frac{1}{x^2+1}-1$:

It seems to be that $g(x)$ seems like $f(x)$, but so as to have been flattened a bit, as if we grabbed it by its two sides and pulled it down onto the x-axis. Is there any way to rigorously define a flattening action in this sense? I'm thinking that surely engineers have to deal with problems related to the malleability of a material under pressure, which would seem to be what is being represented in the comparison of these curves, and so perhaps there could be a way to define this action mathematically.

Answer 1

Nothing rigorous. Just your choice.

Similar to the even function you chose we can choose any other disturbance function $f(x),g(x)$ applied to a parabola that vanishes at the the origin with $y \to 1 \text { as } { x \to \pm \infty}. $ They would all produce same chosen local transient disturbance and same distant offset:

f[x_] = -1/(1 + x^2) + 1
Plot[{x^2, f[x], x^2 - f[x]}, {x, -2.5, 2.5}, GridLines -> Automatic]
g[x_] = -E^(-x^6) + 1
Plot[{x^2, g[x], x^2 - g[x]}, {x, -2.5, 2.5}, GridLines -> Automatic]