Function inside function proofe question

[nhrock3]

it is given that f(x) is monotonically increasing and f(g(x)) is monotonically increasing

does g(x) is monotonically increasing??

i tried to solve it this way:

i think that it does.so i want to disprove the theory that g(x) is not is monotonically increasing:

suppose that g(x) is monotonically decreasing
if a<b then g(a)>g(b)
f(a)<f(b)
f(g(a))<f(g(b))

now what??

 

[michonamona]

Suppose g(x) is monotonically decreasing. Let a<b. Then this means g(a)>g(b). Now since f(g(x)) is monotonically increasing, g(a)>g(b) must imply f(g(a))>f(g(b)). Notice the contradiction with one of our assumption? If not, examine the assumptions we made and the conclusion we ended up with.