Composition of Functions

Below, the purple point traces out the composition: g(f(x)).

This time, take a look at the relationship between the slopes:

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Look at the places where the composition function has horizontal tangents. What must be true of one of the input functions? (Try to formulate the rule specifically.)

What happens where the input functions have negative slopes (at x, and at f(x) )?

What is the apparent relationships between the slopes in question?

Dave Matthews, January 16, 2008, Created with GeoGebra