Generalizing the relation between H(x), F(x) and G(x)

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Discussion Overview

The discussion revolves around the relationship between the functions H(x), F(x), and G(x), specifically exploring whether the equation H(x) = F(x) - G(x) can be generalized based on given conditions at specific points a and b. The scope includes mathematical reasoning and exploration of function properties.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant proposes that if H(x) is defined as the integral of h(x) from c to x, and if H(a) and H(b) equal F(a) - G(a) and F(b) - G(b) respectively, then it might imply H(x) = F(x) - G(x) for all x.
  • Another participant challenges this by asking for examples of functions F and G where the relationship holds at points a and b but fails elsewhere, indicating that the information provided may not be sufficient for a general conclusion.
  • A subsequent reply reiterates the challenge, emphasizing that the relationship H = F - G is only confirmed at x = a and x = b, with no information available for other values of x.
  • One participant suggests that the previous response effectively addresses the initial question, implying a realization about the limitations of the conclusion.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the generalization of the relationship between H, F, and G. The discussion remains unresolved about whether H(x) can be expressed as F(x) - G(x) for all x.

Contextual Notes

The discussion highlights limitations in the provided information, particularly regarding the validity of the relationship outside the specified points a and b. There is an absence of general proof or counterexamples applicable to all x.

andyrk
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If [itex]H(x)= \int_c^x h(x)dx[/itex] and [itex]H(a) = F(a) - G(a) = \int_c^a h(x)dx[/itex] and [itex]H(b) = F(b) - G(b) = \int_c^b h(x)dx[/itex], then does that mean [itex]H(x) = F(x) - G(x)[/itex]? Is the information provided sufficient enough to come to that conclusion?
 
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Can you come up with a sample F and G so that H=F-G holds for x=a and x=b but does not hold in general?
 
Simon Bridge said:
Can you come up with a sample F and G so that H=F-G holds for x=a and x=b but does not hold in general?
No, H = F-G holds only for x = a and x = b. There isn't any information for any x other than a or b.
 
Then you seem to have answered your own question.
 
Haha..yeah..I think I have,
 

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