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.
g(x) = -x
f(x) = -1/x
now g(x) = 1/f(x) = -1/x
g(f(x)) = f(g(x)) = 1/x
also
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f(x) = |1/x| (absolute value function)
g(x) = |x|
now g(x) = 1/f(x) =
g(f(x)) = f(g(x)) = |1/x|
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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
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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
And many more interesting solutions (spoiler?)
