Does it hold: if $f(v+u)=x$ then $v=f^{-1}(x)-u$?

Question

I would like to know if this is true:

$$f(v+u)=x$$ then $$v=f^{-1}(x)-u$$

Thanks a lot in advance.

Answer 1

My assumptions to your question: $f:\Bbb R \rightarrow \Bbb R$ is a injective function (otherwise $f^{-1}(x) $ would be ill-posed). $x,u,v \in \Bbb R$.

So let $x= f(v+u)$. Then $f^{-1}$ is a function on $f(\Bbb R)$ and $$f^{-1}(x) = f^{-1}(f(v+u)) = v + u \\ \Leftrightarrow f^{-1}(x)-u =v $$