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Is the function w(x)=3x7+9x3w(x) = 3x^7 + 9x^3 even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither

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Full solution

Q. Is the function w(x)=3x7+9x3w(x) = 3x^7 + 9x^3 even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither
  1. Substitute and Evaluate: w(x)=3x7+9x3w(x) = 3x^7 + 9x^3\newlineFind w(−x)w(-x) by substituting −x-x for xx.\newlinew(−x)=3(−x)7+9(−x)3w(-x) = 3(-x)^7 + 9(-x)^3
  2. Simplify Expression: w(−x)=3(−x)7+9(−x)3w(-x) = 3(-x)^7 + 9(-x)^3\newlineSimplify the right side.\newlinew(−x)=−3x7−9x3w(-x) = -3x^7 - 9x^3
  3. Compare Functions: Compare w(x)w(x) and w(−x)w(-x).
    w(x)=3x7+9x3w(x) = 3x^7 + 9x^3
    w(−x)=−3x7−9x3w(-x) = -3x^7 - 9x^3
    Since w(−x)=−w(x)w(-x) = -w(x), w(x)w(x) is an odd function.

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