Spring 2017, problem 44

Find two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.


8 months ago

We partition the real axis into a lot of intervals (for two functions the partion can be different.) Let $h(x)=g\circ f(x)$, $k(x)=f\circ g(x)$. First, we define an interval $A$ is low than another $B$ by $A$<$B$, if $\forall a\in A, b\in B$ such that $a$<$b$, like $(-1,2]$ is lower than $(2,4]$. If $h$ is increasing on $A$ and $B$, then we want the image of $A, B$ satisfying $h[A]$<$h[B]$; $k$ is decreasing on $A$ and $B$, then we want the image of $A, B$ satisfying $k[B]$<$k[A]$, which ensures the monotonicity (more) globally. With this idea, we can just arrange the map of partioned intervals.

8 months ago

f(x) = -x(-1)^(floor(|x|)) : http://www.wolframalpha.com/input/?i=-x(-1)%5E(floor(%7Cx%7C))

g(x) = (-x -sgn(x))(-1)^(floor(|x|)) : http://www.wolframalpha.com/input/?i=(-x+-sgn(x))(-1)%5E(floor(%7Cx%7C))

f(g(x)): http://www.wolframalpha.com/input/?i=-((-x-sgn(x))(-1)%5Efloor(modulus(x)))(-1)%5Efloor(modulus((-x-sgn(x))*(-1)%5Efloor(modulus(x))))

g(f(x)): http://www.wolframalpha.com/input/?i=(-(-x(-1)%5E(floor(%7Cx%7C)))+-sgn(-x(-1)%5E(floor(%7Cx%7C))))(-1)%5E(floor(%7C-x(-1)%5E(floor(%7Cx%7C))%7C))

9 months ago

Just sitting on Michel's giant shoulders, try:

f(x) = - sign(x) . sqrt( |x| ) and g(x) = sign (x) . x^2 (with sign(x) = 1 if x > 0, -1 otherwise).

Both f and g are defined on all of R and f o g (x) = - sign (x) .|x| = - x, stricly decreasing and g o f (x) = sign (x) |x| = x, strictly increasing.

In fact this doesn't work by here $g\circ f=f \circ g$,

  • note that here: $f$ decreases, $g$ increases $\Rightarrow~g\circ f$ decreases

  • anyway solutions can't be continous, (otherwise so are $g\circ f, \; f \circ g$ hence 1to1, so are $f,g$ hence strictly monotonous, and $g\circ f, \; f \circ g$, have same monotony).

Hubert 9 months ago
9 months ago

Well if both functions can operate from a subset of R to a subset of R then I would suggest :

f(x) = - sqrt (x) and g(x) = x^2 then f(g(x) = -x strictly decreasing and g(f(x)) = x strictly increasing for x positive otherwise f is not defined. But this sounds too simple...I would love to see an answer with complete mapping of R.

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