All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Reciprocal composition is reciprocal. (Posted on 2014-02-08)
Find, if possible, two functions f and g with:

f(x) ≠ g(x)

g(x) = 1/f(x)

g(f(x)) = 1/f(g(x))

for all x in the respective domains.

 No Solution Yet Submitted by Jer No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 A slightly more interesting solution (spoiler?) | Comment 3 of 4 |
A less trivial offering is
f(x) = x
g(x) = 1/x

Note that
g(x) = 1/f(x) = 1/x
g(f(x)) = f(g(x)) = 1/x

This works as a problem solution if we limit the domain of both to x not in (-1,0,1)

Or, if you prefer, we could transform g(x) into
g(x) = (x^2 - 1)/(x^3 - x).  Arguably, this equals 1/x except for x in (-1,0,1), where it is undefined.

Similarly, f(x) = (x^4 - x^2)/(x^3 - x).
Arguably, this equals x except for x in (-1,0,1), where it is undefined.

f(x) = (x^4 - x^2)/(x^3 - x)
g(x) = (x^2 - 1)/(x^3 - x)
and this seems to meet all puzzle conditions.

It also works if they are reversed:
g(x) = (x^4 - x^2)/(x^3 - x)
f(x) = (x^2 - 1)/(x^3 - x)

 Posted by Steve Herman on 2014-02-08 13:27:57

 Search: Search body:
Forums (0)
Random Problem
Site Statistics
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox: