From my text book it says that $f(x)= x^3$ and $f^{-1}(x) = \sqrt[3]{x}$ , which I totally agree with.
why does $f(x)= \frac 1 {x-1}$ and $f^{-1}(x)= \frac 1 {x + 1}$ and not equal $f^{-1}(x)= \frac 1 {x+1}$?
I know when you inverse a function you reverse the sign values. Can anyone explain this to me a little more thoroughly?
(use curl brackets for multichar objects in FRAC)
I'm not sure what you mean by "you reverse the sign values" - generally you cannot just "tweak" the function slightly in order to get the inverse. If $f(x)=1/(x-1)$ then write down $y=1/(x-1)$, swap the $x$ and $y$, and solve for $y$: $$ x=\frac{1}{y-1} \\ x(y-1)=1 \\ xy=x+1 \\ y=\frac{x+1}{x}=1+\frac{1}{x} $$
You can check your answer by verifying the following: $$f(f^{-1}(x))=x \quad\mbox{and}\quad f^{-1}(f(x))=x$$