Let $P(n)$ be the statement
''$\varphi(x^{-n})=\varphi(x)^{-n}$, ''
and let $\varphi$ be a homomorphism. Consider $P(1)$. Note that \begin{align*} \varphi(x^{-1})=\varphi(x)^{-1} \end{align*} since homomorphisms map inverses to inverses. Suppose $P(k)$ is true for all $k\geq 1$. Observe that \begin{align*} \varphi(x^{-k-1})= \varphi(x^{-k}x^{-1})= \varphi(x^{-k})\varphi(x^{-1})=\varphi(x)^{-k} \varphi(x)^{-1}=\varphi(x)^{-k-1}, \end{align*}
therefore $P(k+1)$ holds. By the principal of mathematical induction, $P(n)$ is true for all $n\in\mathbb{Z}^+$.
I am unsure if this is a valid way to extend the theorem in the title to all integers. I don't think I have violated the principal of induction since I am still only dealing with positive integers in my statement $P(n)$. Any help would be appreciated.