One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. thinking "the functional inverse of $f(x)=x$ 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.

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.