Problem

Determine whether the function \(f(x) = 3x^4 - 2x^2 + 1\) is an odd function, an even function, or neither.

Answer

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Answer

Comparing \(f(-x)\) with \(f(x)\), we find that \(f(-x) = f(x)\), so the function is an even function.

Steps

Step 1 ：An even function is a function for which the following equation holds: \(f(x) = f(-x)\). For an odd function, the following equation should hold: \(f(-x) = -f(x)\).

Step 2 ：So first, let's find \(f(-x)\) for the function \(f(x) = 3x^4 - 2x^2 + 1\).

Step 3 ：\(f(-x) = 3(-x)^4 - 2(-x)^2 + 1 = 3x^4 - 2x^2 + 1\).

Step 4 ：Comparing \(f(-x)\) with \(f(x)\), we find that \(f(-x) = f(x)\), so the function is an even function.

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