What are some common ways students get confused about finding an inverse of a function?

One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. thinking "the inverse of the function $f(x)=x$ is $f^{^{-1}}(x)=\frac{1}{x}$."

I'm inexperienced and I'm trying to anticipate the ways which someone would incorrectly find an inverse and what thinking motivates it.