2024 AMC 12A Problem 13

The curve $y=e^{x+1}+e^{-x}-2$ is symmetric about the vertical line through its minimum. Setting the derivative $e^{x+1}-e^{-x}=0$ gives $x+1=-x,$ so $x=-\tfrac12.$ Reflecting $\left(-1,\tfrac12\right)$ across $x=-\tfrac12$ keeps the $y$-coordinate and sends $x=-1$ to $x=0.$ The image is $\left(0,\tfrac12\right).$ Thus, the correct answer is D.