Idea: Integration by parts.
Hello
Question:
Prove that if f&g are continous and inverse functions for each other and a&b are constant where b>a .. then :
My FAILED try:
Am thinking about a substitution which makes the f(x) be g(x)
So I substitute x=g(g(x)) ..
But this failed; since dx will be compliacted
Any ideas?
I think it works for f increasing...
... but not decreasing...
I.e. subtracting as directed in your formula leaves the white region(s) inside the larger rectangle - which correspond(s) to
in the increasing case only.
Edit: on the other hand...
Ah! Should have tried integration by parts before sounding off...
Spoiler:
Still bothered by my graphs, though...
Late edit: Thanks for exuming this, BayernMunich, but the less said the better! I did spot my VERY SILLY graph error in the end!
Well, come to think of it, f(x) and g(x)--as they are written--cannot, strictly speaking, be inverse functions of each other because they both have the same argument x. In order for two functions to be inverses of each other the one must be the argument of the other and vice versa. I would suggest writing the two functions as y(x) and x(y) and see if that helps to keep track of things.