Linked Questions

2
votes
3answers
1k views

Some examples of functions that are their own inverse? [duplicate]

I'm looking for the name and some examples of functions $f$ with the following property $$f\circ f=I$$ where $I$ is the identity. This means that $f=f^{-1}$; some examples are the functions $f(n)=-n$...
0
votes
1answer
616 views

How many different functions are there that are equal to their own inverse? [duplicate]

I know that functions can be their own inverse such as $f(x)=x$ however I thought there were only two $f(x)=x$ and $f(x)=-x$. Is there more?
4
votes
0answers
70 views

Solving the equation $f(x)=f^{-1}(x)$. [duplicate]

Exactly under what conditions would the equality $f(x)=f^{-1}(x)$ hold? The proof attached below considers a special case when $f$ is strictly increasing. The theorem then says that the set of ...
3
votes
0answers
68 views

$f(x) = f^{-1}(x)$ [duplicate]

I want to find functions that are their own inverse function, so that $$f(x) = f^{-1}(x)$$ and $$f(f(x)) = x$$ I have managed to find the following examples: $$f(x) = x$$ $$f(x) = -x + n$$ $$f(...
39
votes
4answers
15k views

Cardinality of the set of all real functions of real variable

How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?
19
votes
6answers
47k views

Can the inverse of a function be the same as the original function?

I was wondering if the inverse of a function can be the same function. For example when I try to invert $g(x) = 2 - x$ The inverse seems to be the same function. Am I doing something wrong here?
17
votes
2answers
1k views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
7
votes
5answers
5k views

Examples of involutions on $\mathbb{R}$

Just recently I read up something about involutions (functions $f: A \rightarrow A$ such that $f(f(x))=x$, for all $x$ in the domain of $f$), and was wondering how many (if there is a small set of ...
0
votes
3answers
1k views

$f:X\rightarrow X$ such that $f(f(x))=x$

Let $X$ be a metric space and $f:X\rightarrow X$ be such that $f(f(x))=x$, for all $x\in X$. Then $f$ is one-one and onto; is one-one but not onto; is onto but not one-one; need ...
4
votes
1answer
1k views

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$.

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$. I think is should be proved with the help of Cantor-Bernstein theorem. It is easy to ...
0
votes
3answers
79 views

Let $f$ be a continuous function such that $f(f(x))=x$

Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(f(x))=x$. I have to prove or disprove whether $f$ is identity or not. Given conditions imply that $f$ is both injective ...
6
votes
1answer
338 views

How to solve $f(f(x)) = x$?

Seems simple enough, but I have no idea how one would get all solutions to this. Wolfram Alpha gives $5$ answers, the first $2$ of which I could get myself, but the following $3$ completely defeat me.