What is difference between abs(1/x^2) and 1/x^2 ?

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Discussion Overview

The discussion centers on the difference between the mathematical expressions abs(1/x^2) and 1/x^2, exploring their behavior and implications, particularly in the context of real and complex numbers. Participants examine the graphical representations and the effects of absolute values on these functions.

Discussion Character

  • Exploratory, Debate/contested, Conceptual clarification

Main Points Raised

  • Some participants question the difference between abs(1/x^2) and 1/x^2, noting that their graphs appear the same when plotted.
  • Others suggest that while the graphs may be identical for real numbers, the functions themselves are not the same, particularly when considering complex numbers.
  • A participant points out that for complex values, the expressions yield different results, specifically mentioning the case when x is equal to i.
  • Another participant emphasizes that if the graphs are the same, the functions must also be the same, arguing that equality of graphs implies equality of functions.
  • Some participants express uncertainty about the implications of absolute values and encourage further reasoning about what absolute values do in different contexts.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether abs(1/x^2) and 1/x^2 are equivalent, with some asserting they are the same under certain conditions while others highlight differences, especially in complex analysis.

Contextual Notes

Limitations include the dependence on the domain of x (real vs. complex numbers) and the implications of absolute values on the functions being compared.

Md. Abde Mannaf
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what is difference between abs(1/x^2) and 1/x^2 .
i am using https://graphsketch.com/ graph result is same. what do you think?
 
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Md. Abde Mannaf said:
what is difference between abs(1/x^2) and 1/x^2 .
i am using https://graphsketch.com/ graph result is same. what do you think?

Well, what do you think? What does the absolute value do?
 
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Md. Abde Mannaf said:
what is difference between abs(1/x^2) and 1/x^2 .
i am using https://graphsketch.com/ graph result is same. what do you think?
They are both equal. That is, if you allow for only real numbers.

Think what happens for x=i.
abs(1/(i^2)) versus 1/(i^2).
 
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Mentallic said:
Well, what do you think? What does the absolute value do?
i think it is same. function is not same but graph is same. that is my question is are equal?
 
If the graphs are the same, then the functions must be the same. The graph of a function f is the set of all points (x, f(x)). The graph of a function g is the set of all points (x, g(x)). If (x, f(x)) = (x, g(x)) then it is easy to see that you must have f(x) = g(x)
 
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Md. Abde Mannaf said:
i think it is same. function is not same but graph is same. that is my question is are equal?

You can figure it out through some simple reasoning. What does the absolute value do? Give me some examples. And then, what values can't 1/x2 ever be? For example, ##x^2\geq 0## for all values of x, so x2 can't ever be negative.
 
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