Identify the odd function.(1 point)

Question

Responses

f(x)=x4
f left parenthesis x right parenthesis equals x superscript 4 baseline

f(x)=3x2
f left parenthesis x right parenthesis equals 3 x squared

f(x)=2x3
f left parenthesis x right parenthesis equals 2 x cubed

f(x)=2(x+1)3
f left parenthesis x right parenthesis equals 2 left parenthesis x plus 1 right parenthesis cubed

Answer 1

To determine which of the given functions is an odd function, we need to recall the definition of an odd function. A function \(f(x)\) is considered odd if it satisfies the condition:

\[ f(-x) = -f(x) \]

for all \(x\) in its domain.

Let's analyze each function:

1. \(f(x) = x^4\):

   • \(f(-x) = (-x)^4 = x^4\)
   • Since \(f(-x) = f(x)\), this function is even.

2. \(f(x) = 3x^2\):

   • \(f(-x) = 3(-x)^2 = 3x^2\)
   • Since \(f(-x) = f(x)\), this function is even.

3. \(f(x) = 2x^3\):

   • \(f(-x) = 2(-x)^3 = -2x^3\)
   • Since \(f(-x) = -f(x)\), this function is odd.

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

   • \(f(-x) = 2(-x + 1)^3 = 2(1 - x)^3\), which does not simplify to \(-f(x)\).
   • Therefore, this function is neither odd nor even.

Based on this analysis, the odd function among the options is:

\(f(x) = 2x^3\).