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How do you compose two functions, f and g, when the image of the first function...

How do you compose two functions, f and g, when the image of the first function is not a subset of the domain of the second function?

Example: Let g be the map from the Riemann sphere to the complex numbers (where g((0,0,1))=infinity) and f be the conjugation map from the complex numbers to the complex numbers. How can one compute f(g((0,0,1))) when infinity is not a complex number?
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Answer #1

Let the image of the function and the domain of the function

Consider the set

Define the function as,

if otherwise is undefined.

i.e, is not defined if .

This is the definition of composition function.

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