Is the Function f(x) = 6x(1-x) Odd, Even, or Neither?

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Discussion Overview

The discussion centers around determining whether the function f(x) = 6x(1-x) is odd, even, or neither. Participants explore various methods for evaluating the function's properties, including algebraic checks and graphical analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests checking if f(-x) = -f(x) for oddness and f(-x) = f(x) for evenness, noting that the function can also be neither.
  • Another participant proposes plotting the function over a range of x values to visually assess symmetry about the y-axis, indicating that a mirror image suggests evenness, while a negative reflection suggests oddness.
  • A later post asserts that the function is not odd or even, but does not provide justification for this claim.
  • One participant introduces a related question about a transformed version of the function, g(y), and its properties, while clarifying that the evenness or oddness of g(y) does not imply the same for f(x).
  • Another participant emphasizes that a function can be even, odd, neither, or both, reiterating a point made earlier in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the function, with some suggesting it may be neither odd nor even, while others explore methods to determine its properties without reaching a consensus.

Contextual Notes

Some assumptions about the function's behavior may be missing, and the discussion does not resolve the mathematical steps necessary to definitively classify the function.

hofoen
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Is this function odd or even:
f(x) = 6x(1-x)
 
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If it is odd, then f(-x) = -f(x)
If it is even, then f(-x) = f(x).
You can check which one it satisfies.
Note that it can be neither (i.e. if it is not odd, that doesn't necessarily mean it's even).
 
Another approach is, you can plot your function as a graph for different values of x say from -10 to 10. Now you see the graph of f(x) , whether it is symmetrical about y axis. If you see mirror image of function about y-axis it is even function and if you see negative reflection it is odd and if it is neither, then it is neither even nor odd function.
 
hofoen said:
Is this function odd or even.

No, it isn't.
 
Once the OP has settled the answer to his/her original question:

Let x = \frac{1}{2} + y. In terms of y is the function g(y) = f(1/2+y) even or odd?

(Note that this does not really mean that f(x) is even or odd if g(y) is even or odd, since technically speaking f and g are not the same function, even though they are just the translations of each other).
 
Hofoen, please note that a function can be even or odd or neither or both.

That iws uart's point!
 

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