What makes a function quasi-linear?

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Discussion Overview

The discussion revolves around the characterization of a specific function, f = min(1/2, x, x^2), and its classification as quasi-linear. Participants explore the conditions for quasi-linearity, including quasi-convexity and quasi-concavity, and examine the implications of the function's behavior over different intervals.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the quasi-linearity of the function, suggesting it must be both quasi convex and quasi concave.
  • Another participant agrees with the need for both conditions and asserts that the function meets these criteria.
  • There is a discussion regarding the function's concavity, with one participant proposing that the function is not concave on the interval (0,1) because it behaves as x^2, which is convex.
  • A later reply refines this observation, stating that the function is not concave specifically on the interval (0, 1/√2) and suggests an alternative function that would be concave.
  • One participant claims that the function is monotonic and states that every monotonic function is quasi-linear.

Areas of Agreement / Disagreement

Participants express differing views on the concavity of the function and its classification as quasi-linear. While some agree on the function's properties, there is no consensus on the implications of its behavior across different intervals.

Contextual Notes

Participants discuss the function's behavior over specific intervals, but there are unresolved aspects regarding the definitions of quasi-convexity and quasi-concavity as they apply to this function.

newphysist
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Hi,

I have two questions.

(1) I am trying to understand how the following function is quasi-linear:

Code: 
f = min(1/2,x,x^2)

For it to be quasi linear it has to be quasi convex and quasi concave at same time.

(2) I think the reason the above function is not concave is cause on a certain interval (0,1) f = x^2 which is convex. Am I correct in my reasoning?

Thanks guys
 
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newphysist said:
(1) I am trying to understand how the following function is quasi-linear:
Code: 
f = min(1/2,x,x^2)
For it to be quasi linear it has to be quasi convex and quasi concave at same time.
Yes. Which it is.
(2) I think the reason the above function is not concave is cause on a certain interval (0,1) f = x^2 which is convex. Am I correct in my reasoning?
Yes, though it would be more accurate to observe that on (0, 1/√2) it is not concave. (min{.5, x/2, x} would have been concave.)
 
This function is monotonic. And every monotonic function is quasilinear.
 

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