To determine which of the given functions is an odd function, we need to apply the definition of an odd function: A function \( f(x) \) is considered odd if \( f(-x) = -f(x) \) for all \( x \).
Let's evaluate each of the provided functions:
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\( f(x) = 3x^2 \): \[ f(-x) = 3(-x)^2 = 3x^2 = f(x) \] This is an even function, not odd.
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\( f(x) = x^4 \): \[ f(-x) = (-x)^4 = x^4 = f(x) \] This is also an even function, not odd.
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\( f(x) = 2x^3 \): \[ f(-x) = 2(-x)^3 = -2x^3 = -f(x) \] This function is odd.
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\( f(x) = 2(x+1)^3 \): \[ f(-x) = 2(-x+1)^3 = 2(1-x)^3 \quad \text{(needs further simplification)} \] This does not simplify directly to \(-f(x)\), so let’s calculate: \[ f(-x) \neq -f(x) \quad \text{(After evaluating, it's determined that this function is neither odd nor even but not fully evaluated here.)} \]
The only odd function from the ones given is \( f(x) = 2x^3 \).
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