Compositions of Functions

作者：mathtasksproj, Carlos

主題：函數

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 ?

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A
Both and exist.
B
Neither composition exists
C
Only exists
D
Only exists

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b)

The two functions are graphed below, evaluate .

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c)

Can you write a general expression for ? What about ?

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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?

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A
exists but changes must be made for to exist.
B
exists but changes must be made for to exist.
C
Both compositions exist without changes
D
No composition between these functions can exist

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e)

What changes are necessary?

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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."

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