Composition of Functions V2

Given two functions

and

, the composition of the functions

exists if the target of

is a subset of the domain of

. Conversely,

exists if the target of

is a subset of the domain of

. Taking by example the functions

and

, does the composition

exist? What about

? The two functions are graphed below, evaluate

. Can you write a general expression for

? what about

It is not always possible to compose functions because their domains and targets may not match correctly. The next graph shows

and

. Which composition exists without any changes? Which composition has to have the domain/target modified? What changes are necessary?

Examine the next two graphs,

and

;

and

. Is there any intersection between the functions?

Can you make any conjectures about the composition of functions from your observations?

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