MHB Finding Symmetries of a Function

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To find the symmetry of the function g(x) = x^3 - 3, one can replace x with -x and analyze the result; if the original function is returned, it is even, and if the negative is returned, it is odd. Graphing functions, such as y = (x - 3)^2, visually demonstrates symmetry, revealing that this specific function is symmetrical about the line x = 3. There is a discussion on whether all cubic functions exhibit symmetry at a single point rather than along a line. The methods of substitution and graphing are effective for determining symmetry in various functions. Understanding these concepts is crucial for analyzing function behavior in mathematics.
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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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