Question

Let f and g be decreasing functions forall Real numbers. Prove that f o g isincreasing.

Answer 1

Proof with Step-by-step explanation:

Since it is given that f(x) is decreasing thus we infer that

$\frac{d}{dx}f(x)=f'(x)$

Similarly it is given that g(x) is decreasing thus we infer that

$\frac{d}{dx}g(x)=g'(x)$

Now let $y=fog=f[g(x)]$

Differentiating both sides with respect to 'x' and using chain rule of differentiation we get

$dy/dx=\frac{d}{dx}f[g(x)]\\\\y'=f'[g((x)]\times g'(x)$

Now since both f'(x) and g'(x) are both negative(<0) we can infer that their product is always positive (>0)

$\therefore y'(x)>0$

Thus fog(x) is an increasing function.