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Reciprocal composition is reciprocal. (Posted on 2014-02-08) Difficulty: 3 of 5
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    
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Some Thoughts 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.

So, one answer is
    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
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