Is the Fourier transform of the reflection $f(-\cdot)$ equal to the reflection of the Fourier transform?

Question

Suppose that the reflection operator $A:L^2(\mathbb R) \to L^2(\mathbb R)$ is defined as $Af(x)=f(-x)$. Let $F : L^2(\mathbb R) \to L^2(\mathbb R)$ be the Fourier transform. Is it true that $A$ commutes with $F$?