Composition of functions domains

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Discussion Overview

The discussion revolves around the concept of the domains of composite functions, specifically whether the domain of f(g(x)) is always the intersection of the domains of f(x) and g(x). Participants explore different perspectives on how the domains interact based on the range of g and the domain of f.

Discussion Character

  • Conceptual clarification, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants propose that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x).
  • Others argue that it depends on the range of g; if the range of g is contained within the domain of f, then the domain of g determines the domain of f(g(x)).
  • A later reply questions the initial assumption, stating that the domain of f(g(x)) is actually a subset of the domain of g where g(x) falls within the domain of f(x).
  • Another participant suggests that the domain of f(g(x)) can be considered the same as the domain of g, assuming g can accept all real numbers, while acknowledging that f may have a narrower domain.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the relationship between the domains of f, g, and their composition.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the ranges and domains of the functions involved, as well as the implications of these relationships on the overall domain of the composite function.

bcheero
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Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!
 
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It depends on the range of g. If the range of g is contained within the domain of f, then the domain of g is the domain of f(g(x)). If the range of g is not contained, then the part of the range outside is cutoff, inducing a cutoff in the useful domain of of g, i.e. that which ends up as the domain of f(g(x))
 
bcheero said:
Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!
No, it is not always true. In fact it is seldom true. What is true is that the domain of f(g(x)) is that subset of the domain of g such that g(x) is in the domain of f(x).
 
bcheero said:
Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

The domain is simply the set of values your function can accept as input.

The function f o g has the same domain as g which we can pretty safely assume to be all the real numbers.

The domain of f might be more narrow, but f o g has the same domain as g.
 

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