Inverse of sum of two functions

Question

Assuming two functions are invertible, is it true that the inverse of the sum of the two functions is the sum of the inverses (assuming all functions are well behaved)?

Answer 1

As gnometorule pointed in general the sum of invertible is not necessarily invertible.. Anyhow, if two functions are strictly increasing over $\mathbb R$ then their sum is invertible.. Still there is no connection in general between the inverse of the sum and the inverses.

Consider the following simple example:

$$f(X)=X^{5}, g(x)=X \,.$$

Then, both functions are invertible, and so is $f+g$. Anyhow, using Galois Theory, it can be shown that $(f+g)^{-1}(-1)$ is not a number expressible via radicals, let alone in terms of $(-1)$ and $\sqrt[5]{(-1)}=-1$.

Answer 2

No, this one is wrong. The best invertible function is $f(x)=x$. Then $f^{-1}=x$ and $f(x)+f(x)=2f(x)$, but $$(2f(x))^{-1}=\frac{x}{2}$$

Answer 3

Let $f(x) = x, g(x) = -x$, both obviously invertible. Then $(f+g)(x) == 0$, which is not invertible.