TITLE: Domain of $\frac{1}{\frac{1}{x}}$
QUESTION [5 upvotes]: Let $f(x)=\frac{1}{x}$, then we have $f^{-1}(x)=\frac{1}{x}$. So $f(f^{-1})=\frac{1}{\frac{1}{x}}=x$. 
My question is, what is the domain of $f(f^{-1})(x)$? is it everything? or everything but zero?
Then if we plot the graph of $\frac{1}{\frac{1}{x}}$ how will it look like? Will it look just like $y=x$ or $y=x$ with the origin excluded?
I tried to plot using google, the result looks exactly like $y=x$ without the origin removed. But why? I think according to the rule when we compose two functions in this case, the origin must be removed.
Thanks for clarifying my confusion.

REPLY [2 votes]: A quotient is defined when both its numerator and denominator are defined and the denominator is nonzero.
The quotient $\frac1{\frac1x}$ is defined when $1$ is defined (always true) and $\frac1x$ is defined, and $\frac1x$ is nonzero (always true).
The quotient $\frac1x$ is defined when $1$ is defined (always true) and $x$ is defined (always true), and $x$ is nonzero (false when... $x=0$).
Hence, $\frac1x$ is undefined at $x=0$, hence $\frac1{\frac1x}$ is undefined at $x=0$.