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

1. Dec 5, 2011

### NWeid1

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution
I know that it is indeed also increasing, but I'm unsure how to prove it.

2. Dec 5, 2011

### Office_Shredder

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

3. Dec 5, 2011

### NWeid1

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.

4. Dec 5, 2011

### D H

Staff Emeritus
How would you express that in terms of calculus?

5. Dec 5, 2011

### NWeid1

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

6. Dec 5, 2011

### D H

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

7. Dec 5, 2011

### NWeid1

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

8. Dec 5, 2011

### D H

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

9. Dec 5, 2011

### micromass

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.