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)=x^4
f left parenthesis x right parenthesis equals x superscript 4 baseline

f(x)=3x^2

Answer 1

To determine which of the given functions is an odd function, we need to check the property of odd functions: \( f(-x) = -f(x) \) for all \( x \).

Let's evaluate each function:

1. **\( f(x) = 2(x + 1)^3 \)**
Calculate \( f(-x) \):
\[ f(-x) = 2((-x) + 1)^3 = 2(1 - x)^3 \]
This does not equal to \( -f(x) \), as \( f(-x) \neq -f(x) \).

2. **\( f(x) = 2x^3 \)**
Calculate \( f(-x) \):
\[ f(-x) = 2(-x)^3 = 2(-x^3) = -2x^3 \]
This equals \( -f(x) \), so this function is odd.

3. **\( f(x) = x^4 \)**
Calculate \( f(-x) \):
\[ f(-x) = (-x)^4 = x^4 \]
This does not equal \( -f(x) \), as \( f(-x) \neq -f(x) \).

4. **\( f(x) = 3x^2 \)**
Calculate \( f(-x) \):
\[ f(-x) = 3(-x)^2 = 3x^2 \]
This does not equal \( -f(x) \), as \( f(-x) \neq -f(x) \).

Based on these evaluations, the only function that is odd is:

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