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Determine whether the function f(x)=-9x^(2)+1-x^(3) is even, odd or neither. even odd neither

Determine whether the function f(x)=−9x2+1−x3 f(x)=-9 x^{2}+1-x^{3} is even, odd or neither.\newlineeven\newlineodd\newlineneither

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Q. Determine whether the function f(x)=−9x2+1−x3 f(x)=-9 x^{2}+1-x^{3} is even, odd or neither.\newlineeven\newlineodd\newlineneither
  1. Determine Function Type: To determine if the function f(x)f(x) is even, odd, or neither, we need to compare f(x)f(x) with f(−x)f(-x). If f(−x)=f(x)f(-x) = f(x), then the function is even. If f(−x)=−f(x)f(-x) = -f(x), then the function is odd. If neither condition is met, the function is neither even nor odd.
  2. Calculate f(−x)f(-x): First, we calculate f(−x)f(-x) by substituting −x-x for xx in the function f(x)=−9x2+1−x3f(x)=-9x^{2}+1-x^{3}.\newlinef(−x)=−9(−x)2+1−(−x)3f(-x) = -9(-x)^{2} + 1 - (-x)^{3}
  3. Simplify f(−x)f(-x): Simplify the expression for f(−x)f(-x).
    f(−x)=−9x2+1−(−1)x3f(-x) = -9x^{2} + 1 - (-1)x^{3}
    f(−x)=−9x2+1+x3f(-x) = -9x^{2} + 1 + x^{3}
  4. Compare f(−x)f(-x) with f(x)f(x): Now we compare f(−x)f(-x) with f(x)f(x). We have f(x)=−9x2+1−x3f(x) = -9x^{2} + 1 - x^{3} and f(−x)=−9x2+1+x3f(-x) = -9x^{2} + 1 + x^{3}. Since f(−x)f(-x) is not equal to f(x)f(x) and f(−x)f(-x) is not equal to −f(x)-f(x), the function f(x)f(x) is neither even nor odd.

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