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?

Think about the following statement: "The compositions and are always equal, i.e. ." Is it true or false? Why?
To help you test your answer, compare the values of and ; and ; and in the graph below.

The graphs of and below will give you a better idea of why that does not happen. Is there any point where