An Interesting Functional Equation

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[@GhostyOcean]

Immediately I noticed that they're both reciprocals, so f(x)=1/x

[@debasishsarkar5000]

Good equation

[@YouTube_username_not_found]

I remember this problem from an old video of yours on your main channel. "Someone" in the comments gave an explanation for why f(x) = 1/x and f(x) = -1/x are not the only solution.

[@YouTube_username_not_found]

OK, it turned out that that person wrote his explanation in a reply and not a comment. Check under the comment posted by @jpolowin0.

[@YouTube_username_not_found]

An example of a function that solves the equation but is neither f(x) = 1/x nor
f(x) = -1/x *[Disclaimer: it is copy-pasted from the old video]*
Let f(x) = -1/x if and only if x = -1, 1/2, 2, and f(x) = 1/x otherwise. This is a piecewise function that satisfies the functional equation, but is not equal to f(x) = -1/x for all x, and is not equal to f(x) = 1/x for all x.

[@YouTube_username_not_found]

Another example *[Also copy-pasted]*
Here, another example. Let f(x) = 1/x if and only if x = e, 1/(1 - e), 1 - 1/e, π, 1/(1 - π), 1 - 1/π, and f(x) = -1/x otherwise. Again, this satisfies the equation, yet is different from all three of the solutions I have mentioned.

[@YouTube_username_not_found]

It turned out that the person I am mentioning wrote his explanation in a reply, not a comment. Check under the comment of @jpolowin0.

[@YouTube_username_not_found]

It turned out that the person I am mentioning wrote his explanation in a reply, not a comment. Check under the comment of @*******
YT doesn't want me to mention usernames. I already have tried twice and both replies got shadowbanned.

[@renyxadarox]

Multiply (1) and (3) equations first and replace the part of product by the expression of (2) equation. You'll get the same result but faster.

[@mystychief]

Suppose y=1/(1-x) then equation one becomes f(x)f(y)=1/(yx). This suggests f(x)=1/x. Apply in equation one: it fits! Solved.

[@mystychief]

... and of course f(x)=-1/x

[@phill3986]

😊😊😊👍👍👍

[@MrGeorge1896]

f(x) = ±1 / x was easy to guess but hard to proof that it is the only solution.

[@YouTube_username_not_found]

Syber has already posted this problem in an old video on his main channel. Someone in the replies to @jpolowin0 gave an explanation for why those aren't the only solutions. 😀 Go check it out!

[@YouTube_username_not_found]

My replies to you keep getting deleted. ☹

[@YouTube_username_not_found]

Everytime I included the link to Syber's old video or the username of the person I want you to read the replies under his comment, YT decides to obliterate my reply.

[@YouTube_username_not_found]

Syber has already posted this problem in an old video on his main channel. Someone in the replies to @***** gave an explanation for why those aren't the only solutions. Go check it out!

[@YouTube_username_not_found]

Just check my comment to this video and the replies under it.

[@cosmolbfu67]

(1)(3)/(2)
(f(x))^2 = 1/x^2
f(x) = +- 1/x