Algebra Precalculus – Are the Sum and Product of Two Increasing Functions Also Increasing?

Question

Question:

Let $f(x)$ and $g(x)$ be two increasing functions.

a) Show that their sum is also increasing.

b) Investigate the corresponding claim for the product of two increasing functions.

Attempted solution:

a)

If $f(x)$ and $g(x)$ are increasing functions, that means that:

If $x_{1} < x_{2}$, then

$f(x_{1}) < f(x_{2})$

$g(x_{1}) < g(x_{2})$

Adding the two inequalities gives:

$f(x_{1}) + g(x_{1}) < f(x_{2}) + g(x_{2})$

This shows that the sum is also increasing.

b)

Multiplying the original inequalities gives:

$f(x_{1})g(x_{1}) < f(x_{2})g(x_{2})$

This seems false. Even if both are increasing, one of them can be negative, and so the inequality cannot hold for all functions f and g?

Is there a way to formalize this rough intuition-based argument? Or is a counterexample the most obvious way forward? However, a counterexample seems to miss out on the more general argument about why the inequality does not hold for negative functions.

What are some productive ways to finish this question off?

Answer 1

What can be said in general$\,$ is this:

If $f$ and $g$ are two increasing functions on an interval $I$, then:

• If $f(x),\, g(x)>0$ on $I$, $fg$ is increasing.
• If $f(x),\, g(x)<0$ on $I$, $fg$ is decreasing.