How to prove an implication concerning a function $f$ and its inverse?

Question

Does $f(x) = f^{-1}(x)$ imply $f(x) = x$?

I know that the curve of $f$ is symmetrical to that of $f^{-1}$ with the axe of symmetry being $y = x$ but I don't know how to use it in a sound proof of this.

Answer 1

I assume you mean "inverse" and not reciprocal. To answer your question-
No.

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Consider the function $f:\Bbb R \to \Bbb R$ defined as $$f(x) = \begin{cases}\dfrac{1}{x} & x\neq0\\0&x=0\end{cases}.$$

We have that $f = f^{-1}$ but $f(x) = x$ is not true in general.

Answer 2

You will not get a "proof" for this since it is a false statement as you can come up with a counterexample: for example, $f(x)=-x$.

On the other hand, if $f(x)=f^{-1}(x)$, say, for all $x\in\mathbb{R}$. Then you must have $$ f\circ f(x) = f(f(x))=x\;\textrm{for all }x\in\mathbb{R}\;. $$

Such functions are called "involution".