What is the image of 
under the \(B\) function?
No Value composition
Mathematics is a language. It is a tool we use to describe structure. Functions is a part of this language. We use functions to describe relationships, associations, and connections.
We also use functions to describe processes.
We use functions to describe an ordered application of actions. We might see this as step-by-step instructions, an algorithm, or an assembly line.
For example, in a bottling plant the cap is not attached before the beer is poured into the bottle. Beer first, then cap.
For example, the tires are not screwed to the chassi of the car before it is dipped into the rust prevention tank.
For example, Flour, sugar, oil, and eggs are mixed together before baking.
Composition
Let \(F\) and \(G\) be two functions.
The composition of \(F\) and \(G\) is another function built from pieces of \(F\) and \(G\). A circle between the function names denotes a composition.
\[ F \circ G \]
\[ (F \circ G)(a) = F(G(a)) \]
The composition applies \(G\) to an element of the domain of \(G\) and then uses the resulting function value from \(G\) as a domain member of \(F\) obtaining a function value of \(F\).
This ultimate value of \(F\) from this process is the value of \(F \circ G\) at the original domain balue of \(G\).
Composition is our way of representing an ordered application or processes.
Composition
Let \(Shirts\) and \(Pants\) be the two functions represented by the maps below.
Shirts
(arrows go from the domain to the codomain.)
Pants
(arrows go from the domain to the codomain.)
Now define a new function called Dressing.
Dressing = Pants \(\circ \) Shirts
Dressing(
) = (Pants \(\circ \) Shirts)())
Dressing() = Pants(Shirts())
= Pants(
)
= 
Everything has to fit together for a composition to work.
The domain of Dressing is a subset of the domain of Shirts.
The range of Dressing is a subset of the range of Pants.
Composition
Let \(Oufit\) and \(Person\) be the two functions represented by the maps below.
Outfit
(arrows go from the domain to the codomain.)
Person
(arrows go from the domain to the codomain.)
Now define a new function called Dressed.
Dressed = Person \(\circ \) Outfit
Dressed(
) = (Person \(\circ \) Outfit)()
Dressed() = Person(Outfit())
= Person(
)
=
Note: In the previous example, we could not form the composition Outfit \(\circ \) Person, because the values of the function Person are not members of the domain of the function Outfit.
A New Function
The composition of two functions is a new function. It links together two existing functions together to form a brand new function.
- The domain of the composition is a subset of the domain of the first (inner) function.
- The range of the composition is a subset of the range of the second (outer) function.
To evaluate the composition, you make a pitstop in between the domain of the composition and the range of the composition.
But, really, you can ignore the pistop and just think of the composition as its own function.
We are chaining two functions together, which means they must match up.
The range of the inner function has to be inside the domain of the outer function.
If they don’t match up, then we must adjust the domain of the outer function.
Matching
Consider this situation:
We have two functions: \(Inner\) and \(Outer\).
We want to form the composition: \(Outer \circ Inner\).
Suppose \(a\) is in the domain of \(Inner\), but \(Inner(a)\) is not a member of the domain of \(Outer\).
In this case, we need to remove \(a\) from the domain of the composition.
Let \(Inner\) and \(Outer\) be the two functions represented by the maps below.
Inner
(arrows go from the domain to the codomain.)
Outer
(arrows go from the domain to the codomain.)
Notice that \(Inner(\))\( = \)
And, is not in the domain of \(Outer\).
Therefore, cannot be in the domain of the composition.
Now let’s define the composition function, which is a new function called C.
C = Outer \(\circ \) Inner
We must modify the domain.
(arrows go from the domain to the codomain.)
We had to remove both and 
, because they both had as their image in
the \(Inner\) function.
This also affected the range.
Anything paried with is no longer in the range of the composition.
This is very common.
This type of domain and range modification for compositions happens all of the time.
We will encounter this kind of activity every time we look at compositions, which is all the time.
Let \(P\) and \(M\) be the two functions represented by the maps below.
M
(arrows go from the domain to the codomain.)
P
(arrows go from the domain to the codomain.)
Now define \(B = P \circ M\) as the composition function.
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more examples can be found by following this link
More Examples of Functions