P3260
We have
\begin{equation}
\begin{split}
f^{-1}(-B)
& = \{ x \in X : f(x) \in - B \} \\
& = \{ x \in X : f(x) \in - \{ b : b \in B \} \} \\
& = \{ x \in X : f(x) \in \{ - b : b \in B \} \} \\
& = \{ x \in X : \exists \, b \in B \text{ such that } f(x) = - b \} \\
& = \{ x \in X : \exists \, b \in B \text{ such that } - f(x) = b \} \\
& = \{ x \in X : - f(x) \in B \} \\
& = (- f)^{-1}(B)
\end{split}
\end{equation}
$\square$
