# function inside function proofe question

• Oct 13th 2010, 10:48 AM
transgalactic
function inside function proofe question
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??
• Oct 13th 2010, 10:58 AM
Plato
Assuming that the derivatives exist.
We know that $f'(x)>0~\&~[f(g(x))]'=f'(g(x))g'(x)>0$
so what does that imply about $g'(x)?$
• Oct 13th 2010, 11:32 AM
transgalactic
that its positive too
thanks :)