$f$ is strictly increasing and $g$ is decreasing. How to find whether $f\circ g$ and $g\circ f$ are increasing, decreasing, strictly increasing or strictly decreasing?
This is what I did,
$f \circ g=f(g(x))$
If we take $x_1 < x_2$,
$f(x_1) < f(x_2) $ and $\ g(x_1) \ge g(x_2) $
Assuming $f(g(x_1)) < f(g(x_2))$, then,
$g(x_1)<g(x_2) \implies x_2<x_1$
This is a contradiction, therefore our assumption is wrong. After this, what should I assume to prove this? If I assume $f(g(x_1)) \ge f(g(x_2)) \ $, a problem occurs since $f$ is strictly increasing and the assumption has an equality possibility.
Or is there a more effective method than this? Can it be applied to prove the same for $g\circ f$?