Question

if f and g are two increasing function such that fog is defined then fog is
(a) increasing

Answer 1

Given $f,g$ are two increasing functions $⟹f^{\prime }\left(x\right)>0,g^{\prime }\left(x\right)>0$

$fog\left(x\right)=f\left(g\right(x\left)\right)$

Differentiating on both sides

$fo{g}^{\prime }\left(x\right)=f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)$

Let $g\left(x\right)=t$

So $fo{g}^{\prime }\left(x\right)=f^{\prime }\left(t\right)g^{\prime }\left(x\right)$

As $f^{\prime }\left(t\right)>0,g^{\prime }\left(x\right)>0$ ,$fo{g}^{\prime }\left(x\right)>0$

$⟹fog$ is an increasing function