I was asked to prove the result of increasing and decreasing function on the composite function...
prove that:
1- if f(X) inc & g(X) inc, foG inc
2-if F(X) inc & G(X) dec, foG dec
3- if F(X) dec & G(X) inc, fog dec
4- if f(X) dec & G(X) dec, fog inc


please help me in that

i was asked to prove the result of increasing and decreasing function on the composite function...
prove that:
1- if f(X) inc & g(X) inc, foG inc


Look at it this way using the chain rule. We assume that f'(x)>0 and g'(x)>0.

By the chain rule f'(g(x))=f'(g(x))g'(x).

Can you finish now?

Look at it this way using the chain rule. We assume that f'(x)>0 and g'(x)>0.

By the chain rule f'(g(x))=f'(g(x))g'(x).

Can you finish now?.

Hmmm no because I don't want to it by chain rule I mean show only increase and decrease by mathematics... and the increase is not shown by >0... can you find some other way??

Hmmm can u help me in the rest of proves

I'll help yo get started. If f(x) increases, that means that f(x) has a positive slope. For a function with a positive slope f(A+delta_A) > f(A) for delta_A positive. Similarly, f(A-delta_A) < f(A).

Now if g(x) increases, this means that g(B+delta_B) > g(B) for positive delta_B. So let's call g(B) = A, and g(B+delta_B) = A + delta_A. What you have is f(g(B+delta_B)) > f(g(B), hence f(g(x)) increases if both f(x) and g(x) increases.

Conversely, if g(x) decreases then g(B+delta_B)< g(B). So let's call g(B) = A and g(B+delta_B) = A - delta_A. Hence f(g(B+delta_B)) < f(g(B)), and f(g(x)) decreases if f(x) increases and g(x) decreases.

Can you take it from here to prove the last two conditions?

Well, I don't know what you're exactly driving at ebaines, but I find it a little like '+' and '-' signs;

+ and + make +
+ and - make -
- and + make -
- and - make +

Anyway, if I understood well your explanation...

f(x) decrease, and g(x) increase...

f(A+delta_A) < f(A)
So, let f(A) = B and f(A+delta_A) = B - delta_B
Hence, g(f(A+delta_A)) < g(f(A))
And g(f(x)) decreases if f(x) decreases and g(x) increases.

However, I don't even understand what I just wrote. *sigh*:(

Maybe a picture would help. See the attached, which shows the case for f(x) decreasing and g(x) increasing. Starting with g(x) - note that as x increases, g(B_deltB)> g(B). Meanwhile, for f(x) decreasing, f(g(B+deltaB)) < f(g(B)), so f(g(x) decreases. Hope this helps