Let $f : X \to \mathbb{R}$ be a D4364: Real function such that

(i)	$B \subseteq \mathbb{R}$ is a D78: Subset of $\mathbb{R}$

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