Question

Is the function `f(x) = x^6-2x^2+3` even, odd or neither?

Answer 1

`f(x)` can be either of odd or even.

Explanation:

Let n be any even number
Let m be any other even number
(Note m could take on the same value as n)

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Then if `x` is odd we can write `x -> n+1`
Then `(n+1)^2 -> n^2+2n+1`
Then `(n+1)^3 -> n^3 +3n^2 +3n +1`
There will always be a `1 times 1` which when added to all the even numbers (n's) makes the total odd.

If `x` is divisible by 2 then it is even and `x^6` will also be even.

`color(blue)( x^6 " can be either even or odd")`
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If `x^2` is odd then let it be represented by `(n+1)`
We have `2x^2` which is `(n+1) + (n+1)` which becomes `2n+2` which is even.

`color(blue)(2x^2 " is always even")`
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`color(blue)(3 " is always odd")`

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So it boils down to the behaviour of combining 3 and `x^6`

Let `x^6 -> (n+1) -> "odd"`
Let `3 -> (m+1) -> "odd"`

`color(red)("Then "x^6 + 3 -> (n+1) + (m+1) = m+n+2" which is even")`

Likewise if n is even `color(red)("then "x^6 + 3 -> (n) + (m+1) = m+n+1" which is odd")`
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Conclusion:
`f(x)` can be either of odd or even.