<s>[INST] We have a set of positive real numbers $\mathcal{S} = \{n \in \mathbb{R} \mid n > 0\}$.  We also have two functions $f: \mathcal{S} \mapsto \mathbb{R}$ and $g: \mathcal{S} \mapsto \mathbb{R}$ that satisfy $$f(x) + g \left( \frac{1}{x} \right) = g(x) + f \left( \frac{1}{x} \right)$$ for all $x \in \mathcal{S}$.  We can deduce that $$f(x) + f \left( \frac{1}{x} \right) = g(x) + g \left( \frac{1}{x} \right)$$  is true for all $x \in \mathcal{S}$.  We also have **exactly one** value of $k \in \mathbb{R}$ for which $$f(k) = g(k).$$ Define $h(x) = f(x) - g(x)$.
Based on the information given in the problem, we can determine that the value of $k$ is
\begin{itemize}
\item[(A)] 1
\item[(B)] -1
\item[(C)] $\sqrt{2}$
\item[(D)] Any positive real number
\item[(E)] Cannot be determined
\end{itemize}
What is the value of $k$?[/INST]