Compositions 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 .
a)
Take for example the functions , and , does the composition exist? What about ?
b)
The two functions are graphed below, evaluate .
c)
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 .
d)
Which composition exists without any changes? Which composition has to have the domain/target modified?
e)
What changes are necessary?
Examine the next two sets of graphs. The first shows the graphs of and ; the second shows the graphs of and .
Can you write these functions as compositions of simpler functions?
What information about graphs of compositions can you deduce from the pictures below?
Can you make any conjectures about the composition of functions from your observations?
With that in mind, do you think the following statement is true or false? "If we compose two functions f(x) and g(x), we always have , for all values of x."