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

[NWeid1]

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.

Homework Equations

The Attempt at a Solution

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

 

[Office_Shredder]

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

 

[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.

 

[D H]

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?

 

[NWeid1]

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

 

[D H]

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

 

[NWeid1]

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

 

[D H]

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

 

[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.