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Determine whether the function f(x)=8x^(7)-3x^(5)+3x is even, odd or neither. even neither odd

Determine whether the function f(x)=8x7−3x5+3x f(x)=8 x^{7}-3 x^{5}+3 x is even, odd or neither.\newlineeven\newlineneither\newlineodd

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Full solution

Q. Determine whether the function f(x)=8x7−3x5+3x f(x)=8 x^{7}-3 x^{5}+3 x is even, odd or neither.\newlineeven\newlineneither\newlineodd
  1. Define function f(x)f(x): Define the function f(x)f(x). The given function is f(x)=8x7−3x5+3xf(x) = 8x^{7} - 3x^{5} + 3x. To determine if the function is even, odd, or neither, we need to compare f(x)f(x) with f(−x)f(-x).
  2. Calculate f(−x)f(-x): Calculate f(−x)f(-x).\newlineSubstitute −x-x for xx in f(x)=8x7−3x5+3xf(x)=8x^{7}-3x^{5}+3x to get f(−x)f(-x).\newlinef(−x)=8(−x)7−3(−x)5+3(−x)f(-x)=8(-x)^{7}-3(-x)^{5}+3(-x)\newlineNow simplify the right side of the function.\newlinef(−x)=8(−1)7⋅x7−3(−1)5⋅x5+3(−1)⋅xf(-x)=8(-1)^7\cdot x^{7}-3(-1)^5\cdot x^{5}+3(-1)\cdot x\newlinef(−x)=−8x7+3x5−3xf(-x)=-8x^{7}+3x^{5}-3x
  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)=8x7−3x5+3xf(x)=8x^{7}-3x^{5}+3x f(−x)=−8x7+3x5−3xf(-x)=-8x^{7}+3x^{5}-3x 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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