Is $f(x)=\frac{1}{x}$ invertible?

Let $f(x)=\frac{1}{x}$, then $f^{-1}(x)$ must equal $$y=\frac{1}{x}$$ and after swapping the variables $$x=\frac{1}{y}$$and rearranging to solve for $y$, $$\frac{1}{x}=y$$ that being said, can you conclude that $f^{-1}(x)=\frac{1}{x}$? I know it works since, when you take $f(f^{-1}(x))$ the result is $x$, it's just unheard of to me to have a function whose inverse is itself... Any help is appreciated.

• Looks good. Lots of functions equal their own inverse. $f(x)=-x$ for example, but there are more complex ones as well. See this – lulu Oct 3 '17 at 12:56
• Everything correct. – Paolo Leonetti Oct 3 '17 at 12:56
• Every function which is symmetric to the $45^°$ line (i.e. the graph of $f(x)=x$) is its own inverse. Other examples are $f(x)=x$, $f(x)=-x+m$ for arbitrary $m$, $f(x)=\sqrt{1-x^2}$ for $x\in[0,1]$. This is nothing strange, just a geometric observation. – M. Winter Oct 3 '17 at 13:02

This might no longer look strange when you consider that from a geometric point of view, inverting a function is just flippes its graph on the $45^°$-line (i.e. the graph of the function $f(x)=x$).

Now look at the graph of $1/x$, which is perfectly symmetric to this line. No wonder that its inverse is the same. You will immediately see that the graphs of

$$f(x)=x,\qquad f(x)=-x+k,\qquad f(x)=\sqrt{1-x^2}$$

are all symmetric in the same way (on an appropriate domain), hence they are all their own inverses.

• Nice answer! (+1) – R.W Oct 3 '17 at 13:48

Yes, the function is invertible, and is its own inverse. It's also not the only such function, $f(x)=-x$ is another such example.

• Could you show me an example of a function that is not invertible? For reference sake. – joshuaheckroodt Oct 3 '17 at 12:57
• "it's just unheard of to me to have a function whose inverse is itself." There are many functions that are there own inverse. f(x)= 1/x is one, f(x)= x is another, f(x)= -x is yet another. – user247327 Oct 3 '17 at 12:57
• @joshuaheckroodt For example, $f(x)=1$ is not invertible. – Simply Beautiful Art Oct 3 '17 at 12:58
• $f(x)= x^2$ is not invertible over all real numbers though $f(x)= x^2$ with domain "all non-negative real numbers" is invertible. – user247327 Oct 3 '17 at 13:00
• @joshuaheckroodt The function $f(x)=\sin(x)$ is not invertible on $\mathbb R$. – 5xum Oct 3 '17 at 13:01

Actually, there are more examples of functions that are their own inverse: $$f(x)=-x$$ $$f(x)=-\frac1x$$ $$f(x)=x$$ $$f(x)=\begin{cases}x+0.5\text{ if }x-\lfloor x\rfloor<0.5\\x-0.5\text{ otherwise}\end{cases}$$ $$\ldots$$