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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 And many more interesting solutions (spoiler?) Comment 4 of 4 |
g(x) = -x
f(x) = -1/x

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

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


f(x) = |1/x|  (absolute value function)
g(x) = |x|

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


f(x) = 1/x if x is rational
          x if x is irrational
g(x) = x if x is rational
          1/x if x is irrational


All the above, of course, are defined for x not in (0, 1, -1).  Or we can do "transformations" as in the previous post to reform these.

Here, however, is one that works for all x:

f(x) = 2 if x in (0,1,-1)
      = x otherwise

g(x) = 1/2 if x in (0, 1 -1)
       = 1/x otherwise

Obviously, lots of other piecewise functions are available 


  Posted by Steve Herman on 2014-02-08 13:56:19
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