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Determine whether the function f(x)=-x^(7)-x+x^(3) is even, odd or neither. neither even odd

Determine whether the function f(x)=−x7−x+x3 f(x)=-x^{7}-x+x^{3} is even, odd or neither.\newlineneither\newlineeven\newlineodd

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Full solution

Q. Determine whether the function f(x)=−x7−x+x3 f(x)=-x^{7}-x+x^{3} is even, odd or neither.\newlineneither\newlineeven\newlineodd
  1. Define function f(x)f(x): Define the function f(x)f(x). The given function is f(x)=−x7−x+x3f(x) = -x^7 - x + x^3.
  2. Determine f(−x)f(-x): Determine f(−x)f(-x).\newlineSubstitute −x-x for xx in f(x)f(x) to find f(−x)f(-x).\newlinef(−x)=−(−x)7−(−x)+(−x)3f(-x)=-(-x)^7-(-x)+(-x)^3
  3. Simplify f(−x)f(-x): Simplify f(−x)f(-x).\newlineSimplify the expression for f(−x)f(-x) by calculating the powers of −x-x.\newlinef(−x)=−(−x)7+x−(−x)3f(-x)=-(-x)^7+ x -(-x)^3\newlinef(−x)=−(−1)7⋅x7+x−(−1)3⋅x3f(-x)=-(-1)^7\cdot x^7+ x -(-1)^3\cdot x^3\newlinef(−x)=−(−1)⋅x7+x−(−1)⋅x3f(-x)=-(-1)\cdot x^7+ x -(-1)\cdot x^3\newlinef(−x)=x7+x−x3f(-x)=x^7+ x -x^3
  4. Compare f(x)f(x) and f(−x)f(-x): Compare f(x)f(x) and f(−x)f(-x). We have the original function f(x)=−x7−x+x3f(x) = -x^7 - x + x^3 and the transformed function f(−x)=x7+x−x3f(-x) = x^7 + x - x^3. Since f(−x)f(-x) is not equal to f(x)f(x) and f(−x)f(-x) is also not equal to −f(x)-f(x), the function f(x)f(x) is neither even nor odd.

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