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Is the function q(x)=x6−9q(x) = x^6 - 9 even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither

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Full solution

Q. Is the function q(x)=x6−9q(x) = x^6 - 9 even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither
  1. Check Even Function: Check if q(x)q(x) is even.\newlineAn even function satisfies q(x)=q(−x)q(x) = q(-x).\newlineCalculate q(−x)q(-x) by substituting −x-x for xx in q(x)=x6−9q(x) = x^6 - 9.\newlineq(−x)=(−x)6−9q(-x) = (-x)^6 - 9.
  2. Calculate q(−x)q(-x): Simplify q(−x)q(-x).\newlineSince (−x)6=x6(-x)^6 = x^6 (because the exponent is even), we get:\newlineq(−x)=x6−9q(-x) = x^6 - 9.
  3. Simplify q(−x)q(-x): Compare q(x)q(x) and q(−x)q(-x). We have q(x)=x6−9q(x) = x^6 - 9 and q(−x)=(−x)6−9q(-x) = (-x)^6 - 9. Since q(x)=q(−x)q(x) = q(-x), the function q(x)q(x) is even.

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