Is gof^-1 = fog?
If f: A → B, g: B → C are functions then define gof and fog.
I am so confused about fog, I want to know the domain and co-domain of fog.

To clarify the relationship between the compositions of functions, let's break down the concepts of \( g \circ f \) (read as "g of f") and \( f \circ g \) (read as "f of g"). These notations represent function compositions, and understanding their definitions and properties will help clear up your confusion.

Defining Function Composition

When we have two functions, \( f: A \to B \) and \( g: B \to C \), we can compose them in two ways:

• g of f (g ∘ f): This means we first apply the function \( f \) and then apply the function \( g \) to the result of \( f \).
• f of g (f ∘ g): This means we first apply the function \( g \) and then apply the function \( f \) to the result of \( g \).

Understanding g ∘ f

The composition \( g \circ f \) is defined as follows:

• Take an element \( x \) from the domain \( A \).
• Apply \( f \) to \( x \) to get \( f(x) \), which will be in the codomain \( B \).
• Then apply \( g \) to \( f(x) \) to get \( g(f(x)) \), which will be in the codomain \( C \).

So, the composition \( g \circ f: A \to C \) is a function that takes an input from \( A \) and produces an output in \( C \).

Exploring f ∘ g

Now, let’s look at \( f \circ g \):

• Take an element \( y \) from the domain \( B \).
• Apply \( g \) to \( y \) to get \( g(y) \), which will be in the codomain \( C \).
• Then apply \( f \) to \( g(y) \). However, this step is only valid if \( g(y) \) is in the domain of \( f \), which it typically is not, since \( f \) maps from \( A \) to \( B \).

Thus, \( f \circ g \) is generally not defined unless the output of \( g \) falls within the domain of \( f \). Therefore, \( f \circ g \) does not have a standard form like \( g \circ f \) does.

Domains and Codomains

Now, let’s clarify the domains and codomains for both compositions:

• Domain of g ∘ f: The domain is \( A \), the set from which \( f \) takes its inputs.
• Codomain of g ∘ f: The codomain is \( C \), the set to which \( g \) maps its outputs.
• Domain of f ∘ g: The domain is \( B \), the set from which \( g \) takes its inputs.
• Codomain of f ∘ g: The codomain is \( B \) (if defined), but this composition is often not valid unless specific conditions are met.

Key Takeaways

In summary, \( g \circ f \) is a valid composition that takes inputs from \( A \) and produces outputs in \( C \). On the other hand, \( f \circ g \) is generally not valid unless the output of \( g \) is in the domain of \( f \). Therefore, \( g \circ f \) is not equal to \( f \circ g \) in most cases, and understanding their definitions helps clarify their distinct roles in function composition.