Composition of functions in inequalities

Question

i'm doing a demonstration, and i am stuck in this part. If i have an inequality such as:

$f(x)\leq g(x)$

Can i compose both sides? Like this, for example:

$h(f(x))\leq h(g(x))$

What would happen if $h(x)$ is monotonically decreasing? I think it would change the inequality, but I'm not sure.

Answer 1

In fact, I would say that this doesn't really have to do with composition of functions at all. Suppose that $v$ and $w$ are real numbers with $v\le w$.

• If $h$ is increasing, then we can conclude that $h(v) \le h(w)$.
• If $h$ is decreasing, then we can conclude that $h(v) \ge h(w)$.
• If neither, then we can't immediately draw any conclusion.

All of this holds with $v=f(x)$ and $w=g(x)$ just the same, but it doesn't really have to do with the fact that $v$ and $w$ are values of two functions at the same point.

Answer 2

This is true of $h$ is increasing. If $h$ is decreasing, the inequality would flip. If $h$ is not monotonic, you cannot compare the compositions.