Inverse of the exponential map in local coordinates

Question

Let $(M,g)$ be a Riemannian manifold and $(\varphi,U)$ a local coordinate patch (not necessarily exponential coordinates) so that $x,y \in U$. Does the following expression hold in local coordinates when it is well defined after identifying $T_yM$ with $\mathbb{R}^n$? $$x-y = \exp_y^{-1}(x)$$

Answer 1

No, unless $\varphi = \exp_y$ (up to translation).

You can assume that $y=0$ (i.e. $\varphi(y) = 0$) after translating $\varphi$. Then the question is: does $x = {\exp_y}^{-1}(x)$ hold for every $x$ in $U$? That's just saying that $\varphi = \exp_y$.