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?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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