# If f and g are monotonic, is f(g(x))?

## Homework Statement

If f and g are both increasing functions, is it true that f(g(x)) is also increasing? Either prove that it is true or five an example that proves it false.

## The Attempt at a Solution

I know that it is indeed also increasing, but I'm unsure how to prove it.

Office_Shredder
Staff Emeritus
Gold Member
It might help to write down the definition of an increasing function

A function f(x) is increasing at a point x0 if and only if there exists some interval containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in the interval to the right of x0.

D H
Staff Emeritus
A function f(x) is increasing at a point x0 if and only if there exists some interval containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in the interval to the right of x0.
How would you express that in terms of calculus?

Um idk by saying that as x increases, so does the y values.

D H
Staff Emeritus
Try it without words. Write an equation. Using calculus.

If x>y then f(x)>f(y)?

D H
Staff Emeritus
You posted this question in "Calculus and beyond," so use calculus.

You don't really need calculus here.

Just use that

$$x<y~\Rightarrow~f(x)<f(y)$$

and

$$x<y~\Rightarrow~g(x)<g(y)$$

Put these two things together.