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Determine whether the function f(x)=-x^(3)+x^(5) is even, odd or neither. even odd neither

Determine whether the function f(x)=−x3+x5 f(x)=-x^{3}+x^{5} is even, odd or neither.\newlineeven\newlineodd\newlineneither

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Full solution

Q. Determine whether the function f(x)=−x3+x5 f(x)=-x^{3}+x^{5} is even, odd or neither.\newlineeven\newlineodd\newlineneither
  1. Evaluate f(−x)f(-x): Consider the function f(x)=−x3+x5f(x)=-x^{3}+x^{5}. To determine if the function is even, odd, or neither, we need to evaluate f(−x)f(-x) and compare it to f(x)f(x). Substitute −x-x for xx in f(x)=−x3+x5f(x)=-x^{3}+x^{5}. f(−x)=−(−x)3+(−x)5f(-x)=-(-x)^{3}+(-x)^{5}
  2. Simplify function: Simplify the right side of the function.\newlinef(−x)=−(−x3)+(−x5)f(-x)=-(-x^3)+(-x^5)\newlinef(−x)=−(−x×x×x)+(−x×x×x×x×x)f(-x)=-(-x\times x\times x)+(-x\times x\times x\times x\times x)\newlinef(−x)=−(x3)+(−x5)f(-x)=-(x^3)+(-x^5)\newlinef(−x)=x3−x5f(-x)=x^3-x^5
  3. Compare f(−x)f(-x) with f(x)f(x): Compare f(−x)f(-x) with f(x)f(x). We have f(x)=−x3+x5f(x)=-x^3+x^5 and f(−x)=x3−x5f(-x)=x^3-x^5. Since f(−x)=−f(x)f(-x) = -f(x), f(x)f(x) is an odd function.

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