AMC 12 2024a Problem 13

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?

$\textbf{(A) }\left(-1,-\frac{3}{2}\right) \textbf{(B) }(-1,0) \textbf{(C) }\left(-1,\tfrac{1}{2}\right) \textbf{(D) }\left(0,\frac{1}{2}\right) \textbf{(E) }\left(3,\frac{1}{2}\right)$