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Is the function k(x)=−5x7−2x3+xk(x) = -5x^7 - 2x^3 + x even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither

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Full solution

Q. Is the function k(x)=−5x7−2x3+xk(x) = -5x^7 - 2x^3 + x even, odd, or neither?\newlineChoices:\newline(A)even\newline(B)odd\newline(C)neither
  1. Check Even Function: Check if k(x)k(x) is even by evaluating k(−x)k(-x) and comparing it to k(x)k(x).k(−x)=−5(−x)7−2(−x)3+(−x)k(-x) = -5(-x)^7 - 2(-x)^3 + (-x)
  2. Simplify k(−x)k(-x): Simplify k(−x)k(-x).
    k(−x)=−5(−x)7−2(−x)3−xk(-x) = -5(-x)^7 - 2(-x)^3 - x
    k(−x)=5x7+2x3−xk(-x) = 5x^7 + 2x^3 - x
  3. Compare k(−x)k(-x) with k(x)k(x): Compare k(−x)k(-x) with k(x)k(x).k(x)=−5x7−2x3+xk(x) = -5x^7 - 2x^3 + xk(−x)=5x7+2x3−xk(-x) = 5x^7 + 2x^3 - xSince k(−x)k(-x) is not equal to k(x)k(x), k(x)k(x) is not even.
  4. Check Odd Function: Check if k(x)k(x) is odd by evaluating if k(−x)k(-x) is equal to −k(x)-k(x).−k(x)=−(−5x7−2x3+x)-k(x) = -(-5x^7 - 2x^3 + x)
  5. Simplify −k(x)-k(x): Simplify −k(x)-k(x).−k(x)=5x7+2x3−x-k(x) = 5x^7 + 2x^3 - x
  6. Compare −k(x)-k(x) with k(−x)k(-x): Compare −k(x)-k(x) with k(−x)k(-x).
    −k(x)=5x7+2x3−x-k(x) = 5x^7 + 2x^3 - x
    k(−x)=5x7+2x3−xk(-x) = 5x^7 + 2x^3 - x
    Since −k(x)-k(x) is equal to k(−x)k(-x), k(x)k(x) is odd.

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