Composition of Functions
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