Composite Functions: Inverse of f o g?

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Discussion Overview

The discussion revolves around the properties of composite functions, specifically whether the inverse of the composition of two functions, f o g, can be expressed as gof. Participants explore the implications of function composition and the conditions under which inverses exist.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether (f o g) ^-1 = gof is true and suggests that it may not be the case.
  • Another participant argues that the notation f o g does not represent a function in the way the original question implies, emphasizing that functions are read from right to left.
  • A request for a counterexample is made, highlighting the need for clarification on the conditions under which the original statement holds.
  • Further clarification is provided that a counterexample would involve a function g whose image does not lie within the domain of f, indicating a potential misunderstanding of function composition.
  • One participant expresses skepticism about the original question being homework-related, suggesting that the inquirer should be able to find a counterexample based on the reasoning provided.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the original statement regarding the inverse of composite functions. Multiple competing views are presented regarding the nature of function composition and the existence of inverses.

Contextual Notes

The discussion highlights limitations related to the definitions of functions and the conditions necessary for the existence of inverses, which remain unresolved.

Monsu
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If f:A -> B and g:B -> C are functions, is this true: f o g is also a function and (f o g) ^-1 = gof

I think this isn't true, but if this isn't the case, could someone please tell me a counter example?? Thanks
 
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with that notation fog is not a function, gof is a function from A (to B thence) to C. but that is just the notational convention: functions read from right ot left. the inverse part *is* wrong. firstly that isn't function, never mind one that posses an inverse (which a function) may or may not do. So it's hard to find a counter example given that.
 
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Thanks, but what could be a counterexample for it??
 
erm, counter example to what? g sends an element of B to an element in C, right? You cannot then follow that with a function from A to anywhere since C is not necessarily a subset of A. counter example is any function g where its image does not lie in the domain of f.

this looks a lot like homework, so i think you should be able to find a counter example if you want to; I've given you the reason why it is not necessairly true
 
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