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Is the function r(x)=2x4−6x2+8xr(x) = 2x^4 - 6x^2 + 8x even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither

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Q. Is the function r(x)=2x4−6x2+8xr(x) = 2x^4 - 6x^2 + 8x even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither
  1. Check even function: Check if r(x)r(x) is even by substituting −x-x for xx and comparing r(−x)r(-x) to r(x)r(x).r(−x)=2(−x)4−6(−x)2+8(−x)r(-x) = 2(-x)^4 - 6(-x)^2 + 8(-x)
  2. Simplify r(−x)r(-x): Simplify the expression for r(−x)r(-x).r(−x)=2x4−6x2−8xr(-x) = 2x^4 - 6x^2 - 8x
  3. Compare r(−x)r(-x) with r(x)r(x): Compare r(−x)r(-x) with r(x)r(x).
    r(x)=2x4−6x2+8xr(x) = 2x^4 - 6x^2 + 8x
    r(−x)=2x4−6x2−8xr(-x) = 2x^4 - 6x^2 - 8x
    Since r(−x)≠r(x)r(-x) \neq r(x), r(x)r(x) is not even.
  4. Check odd function: Check if r(x)r(x) is odd by checking if r(−x)=−r(x)r(-x) = -r(x).−r(x)=−2x4+6x2−8x-r(x) = -2x^4 + 6x^2 - 8x
  5. Compare −r(x)-r(x) with r(−x)r(-x): Compare −r(x)-r(x) with r(−x)r(-x).
    −r(x)=−2x4+6x2−8x-r(x) = -2x^4 + 6x^2 - 8x
    r(−x)=2x4−6x2−8xr(-x) = 2x^4 - 6x^2 - 8x
    Since −r(x)≠r(−x)-r(x) \neq r(-x), r(x)r(x) is not odd.
  6. Conclude function type: Conclude whether r(x)r(x) is even, odd, or neither. Since r(x)r(x) is neither even nor odd, the correct choice is (C) neither.

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