Finding Symmetries of a Function

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    Function Symmetries
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SUMMARY

The discussion focuses on determining the symmetry of functions, specifically the cubic function g(x) = x^3 - 3. To identify symmetry, one can substitute x with -x; if the original function is returned, it is even, and if the negative of the original function is obtained, it is odd. Additionally, graphing techniques are highlighted, with the example of y = (x - 3)^2 demonstrating symmetry about the line x = 3. The conversation concludes with the assertion that cubic functions are symmetrical at a single point rather than along a line.

PREREQUISITES
  • Understanding of function symmetry concepts
  • Familiarity with cubic functions
  • Basic graphing techniques
  • Knowledge of even and odd functions
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  • Explore the properties of even and odd functions in detail
  • Learn about graphing techniques for identifying function symmetries
  • Study the characteristics of cubic functions and their graphs
  • Investigate advanced symmetry concepts in higher-degree polynomials
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Mathematics students, educators, and anyone interested in understanding function symmetries and their graphical representations.

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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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