Question

Identify the odd function.(1 point)
Responses

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

f(x)=2x^3
f left parenthesis x right parenthesis equals 2 x cubed

f(x)=3x^2
f left parenthesis x right parenthesis equals 3 x squared

f(x)=x^4

Answer 1

To identify the odd function among the given options, we need to recall that a function \( f(x) \) is considered odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \).

Let's evaluate each function:

1. \( f(x) = 2(x + 1)^3 \)
- \( f(-x) = 2(-x + 1)^3 = 2(-1)(x - 1)^3 = -2(x + 1)^3 \) (not equal to \(-f(x)\), hence not odd)

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

3. \( f(x) = 3x^2 \)
- \( f(-x) = 3(-x)^2 = 3x^2 = f(x) \) (this function is even, not odd)

4. \( f(x) = x^4 \)
- \( f(-x) = (-x)^4 = x^4 = f(x) \) (this function is also even, not odd)

Based on this analysis, the only odd function among the choices is:

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