Question

Identify the odd function.(1 point)
Responses

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

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

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

f(x)=3x2

Answer 1

To determine which function is an odd function, we need to check if the function satisfies the property of odd functions, which is:

\( f(-x) = -f(x) \)

We'll evaluate each function:

1. **\( f(x) = x^4 \)**:
- \( f(-x) = (-x)^4 = x^4 \)
- Since \( f(-x) \neq -f(x) \), this is not an odd function.

2. **\( f(x) = 2(x + 1)^3 \)**:
- \( f(-x) = 2(-x + 1)^3 = 2(-x + 1)^3 = 2(-1)(x - 1)^3 = -2(x - 1)^3 \) which does not equal \( -f(x) \).
- Therefore, this is not an odd function.

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

4. **\( f(x) = 3x^2 \)**:
- \( f(-x) = 3(-x)^2 = 3x^2 \)
- Since \( f(-x) \neq -f(x) \), this is not an odd function.

Thus, the odd function among the responses is:

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