A light spring with spring constant k has one end attached to a...

The work done by the spring force is given by $W = \dfrac{1}{2}kx_i^2 - \dfrac{1}{2}kx_f^2$, where $x_i$ and $x_f$ are the initial and final deformations (elongation or compression) of the spring from its natural length.

Given that the work done by the spring is $W = \dfrac{1}{2}kx^2$, we have:

$\dfrac{1}{2}kx_i^2 - \dfrac{1}{2}kx_f^2 = \dfrac{1}{2}kx^2$

For the spring initially stretched by a distance $x$ and finally returning to its natural length, $x_i = x$ and $x_f = 0$. Substituting these values, we get $W = \dfrac{1}{2}kx^2 - 0 = \dfrac{1}{2}kx^2$.

For the spring initially compressed by a distance $x$ and finally returning to its natural length, $x_i = -x$ and $x_f = 0$. Substituting these values, we get $W = \dfrac{1}{2}k(-x)^2 - 0 = \dfrac{1}{2}kx^2$.

For the spring initially at its natural length and finally reaching a compressed or stretched position by a distance $x$, $x_i = 0$ and $x_f = \pm x$. The work done is $W = 0 - \dfrac{1}{2}kx^2 = -\dfrac{1}{2}kx^2$.

Thus, the situations where the spring was initially stretched or compressed by a distance $x$ and finally returned to its natural length are the correct ones.