2024 AMC 12A 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)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.