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

New Resources