(b) We start by considering $(F^2f)(x) = F(Ff)(x)$.

We have that: \begin{equation}

F(f(x)) = \hat{f}(k) = \frac{1}{\sqrt{2\pi{}}}\int_{-\infty}^{\infty} f(x)e^{-ikx}dx \end{equation}

Now consider that: \begin{equation}

f(x) = \frac{1}{\sqrt{2\pi{}}}\int_{-\infty}^{\infty} \hat{f}(k)e^{ikx}dk \end{equation}

Then \begin{equation}

F^2(f(x)) = F(F(f(x)) = F(\hat{f}(k)) = \frac{1}{\sqrt{2\pi{}}}\int_{-\infty}^{\infty} \hat{f}(k)e^{-ikx}dk = f(-x) \end{equation}

Since an even function is defined such that $f(-x) = f(x)$, the above operation will recover the original function. For an odd function, which has the property that $f(-x) = -f(x)$, the operation will recover the negative of the function.