Something brought up in class.

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Discussion Overview

The discussion revolves around the composition of functions, specifically the notation and implications of composing two functions f and g, where f: A -> B and g: B -> C. Participants explore the conditions under which compositions like g*f and f*g can be defined, as well as the potential confusion arising from different notational conventions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that while g*f is valid, the composition f*g raises questions about its existence due to the domains and ranges of the functions involved.
  • Another participant challenges the assertion that f(x) cannot equal anything, arguing that it is sufficient for x to be in A for f(x) to be defined.
  • There is a proposal to explore conditions under which f*g could exist, despite the general assertion that it does not.
  • Participants note the importance of clarifying notation, as f*g can mean different things depending on the context, with some suggesting it is common to interpret it as g composed with f.
  • One participant introduces concepts from algebra, discussing the image and kernel of functions, and how these relate to function composition.
  • A suggestion is made that the discussion might relate to inverse functions, particularly if both f and g are invertible.
  • Clarification on the notation used for composition is sought by one participant, indicating a need for confirmation on the teacher's preferred notation.

Areas of Agreement / Disagreement

Participants express differing views on the validity and interpretation of function composition, with no consensus reached on the conditions under which f*g can be defined or the implications of different notational conventions.

Contextual Notes

Limitations include potential misunderstandings of notation and the need for clarity on definitions of function composition. The discussion also reflects varying levels of comfort with the concepts being debated.

JasonRox
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I should have mentionned it, but I rather think about it first.

Suppose f:A->B and g:B->C

then g*f:A->C

That makes sense.

Then they talked about f*g!

Is this possible? Think about it. If f has domain A, and g has range C, than how can f(x) equal anything, since the range of C is not equal to set A, which is what we need to calculate f.

f*g:B->?

I hope you understand.
 
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no that definitely doesn't make sense. i think you're right, but you also shouldn't rock the boat. just do whatever it takes to get through the course.
 
how can f(x) equal anything, since the range of C is not equal to set A, which is what we need to calculate f.

That's not quite true; all we need is for x to be in A.


Although f*g certainly will not exist in general, can you come up with conditions for which it would?


edit: I'm assuming you're using the * symbol to mean composition
 
Last edited:
well one needs to define upper star. for instance the usual definition of upper star is "precede by" so f*g would actually mean gof which is defined as above.
 
no that definitely doesn't make sense. i think you're right, but you also shouldn't rock the boat. just do whatever it takes to get through the course.

IMHO this is, in general, terrible advice. Asking questions is how you learn things (and sometimes can help other people learn things, because they were too shy to ask a question). The bar is so low that just "doing what it takes" will certainly harm your chances in future math classes in which you're expected to know this stuff learned in this class.
 
There are two ways to write composition of functions, check which one your teacher uses. writing f*g to mean g composed with f is very common especially amongst logicians, who see it as more logical, and some applied people who read only read left to right.
 
In algebra the composition leads to the concept of image and kernel. If f: A -> B, then f(a) is the image of A in B under f; the set of points in B that are images of points in A under the function f. And in algebra we have a zero element (group identity or whatever). Kernel f is the set of points in A that are mapped into the zero or identity element of B.
 
I'm wondering if this wasn't supposed to be about the inverse functions.

If f:A->B and g:B->C then g*f:A->C.

If f and g are both invertible, then so is g*f and (g*f)-1= f-1*g-1:C->A.
 
I meant * as a composition.

I will ask to make sure.
 

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