Showing that $f(a)=\Vert p+q\Vert-\Vert p- q\Vert$ has one root, where $p:=(x+a y)/\Vert x+ay\Vert$ and $q:=(x-ay)/\Vert x-ay\Vert$

Question

Let $X$ be a normed linear space and $ x,y \in X\setminus \{ 0\} $. Consider the function $f: \mathbb{R} \rightarrow \mathbb{R} $ defined by \begin{align*} f(\alpha)= \bigg\Vert \frac{x + \alpha y}{\Vert x + \alpha y \Vert} + \frac{ x - \alpha y}{ \Vert x - \alpha y \Vert} \bigg\Vert -\bigg \Vert \frac{x + \alpha y}{\Vert x + \alpha y \Vert} - \frac{ x - \alpha y}{ \Vert x - \alpha y \Vert} \bigg \Vert \quad (\alpha >0). \end{align*} How we can show that this function has exactly one root? Of course we know that $ f(0)>0 $ and there exists $ \alpha \in \mathbb{R} $ such that $ f(\alpha)<0 $. In fact, we guess that the function is strictly decreasing but we have not been able to prove it.