Why is the graph of the reciprocal function $f(x)=\frac{1}{x}$ not a one-to-one function?

Question

The horizontal line test seems to show that no horizontal line intersects the graph of $f$ at more than one point. However, the textbook tells me it's not one-to-one. It is related to the asymptotes, I believe, but I'm not sure how.

Answer 1

If the domain of $f$ is taken to be the entire real line $\mathbb{R}$, then $f(0)$ is not defined. Likewise there is no value of $x$ that gives $f(x)=0$.

However, $f(x)$ is one-to-one if the domain is restricted to $\mathbb{R}\setminus\{0\}$.

ETA: In fact, on $f$'s domain $\mathbb{R}\setminus \{0\}$, not only if $f$ one-to-one, but it is its own inverse!

Answer 2

The textbook might have some nitpick in mind like "$f$ is not a one-to-one function because it's not a function: it's not defined at $0$".

But the horizontal line test tells us that there can't be two inputs which $f$ sends to the same output. This means if we restrict our domain to anything where $f$ is a function (such as the nonzero real numbers) then $f$ will be one-to-one. So you're basically right.