Finding Symmetries of a Function

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    Function Symmetries
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Discussion Overview

The discussion revolves around finding the symmetries of functions, particularly focusing on the function \( g(x) = x^3 - 3 \). Participants explore various methods for identifying symmetries, including algebraic substitutions and graphical analysis.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant suggests replacing \( x \) with \( -x \) to determine if a function is even or odd, noting that if the original function is returned, it is even, and if the negative is returned, it is odd.
  • Another participant mentions that graphing can help visualize symmetries, providing the example of the function \( y = (x-3)^2 \) which is symmetrical about the line \( x = 3 \).
  • A later reply reiterates the previous methods and questions whether all cubic functions are symmetrical at a single point rather than along a line.

Areas of Agreement / Disagreement

Participants express different methods for finding symmetries, but there is no consensus on whether all cubic functions exhibit symmetry at a single point or along a line.

Contextual Notes

The discussion does not clarify the definitions of symmetry being used or the implications of the proposed methods on different types of functions.

megacat8921
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How would one find the symmetry of the function x^3-3=g(x) ? Or any other symmetry.
 
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I would replace $x$ with $-x$, and see what happens. Do you get the original back at you? Then it's even. Do you get the negative? Then it's odd.
 
Graphing is another method. For example, graphing $$y=(x-3)^2$$ makes it easy to see the function is symmetrical about the line $x=3$.
 
Ackbach said:
I would replace $x$ with $-x$, and see what happens. Do you get the original back at you? Then it's even. Do you get the negative? Then it's odd.

greg1313 said:
Graphing is another method. For example, graphing $$y=(x-3)^2$$ makes it easy to see the function is symmetrical about the line $x=3$.

Can we say that all cubic functions are always symmetrical at a single point, and never along a line?
 

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