Question

Identify the odd funtion.

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

Answer 1

2. be correct

Answer 2

To determine whether a function is odd, we need to check if \( f(-x) = -f(x) \) for all \( x \). Let's evaluate each function:

1. **\( f(x) = 2x^3 \)**
- \( f(-x) = 2(-x)^3 = 2(-x^3) = -2x^3 = -f(x) \)
- This function is odd.

2. **\( f(x) = 2(x + 1)^3 \)**
- \( f(-x) = 2(-x + 1)^3 = 2(1 - x)^3 \)
- Now, \( (1 - x)^3 = 1 - 3x + 3x^2 - x^3 \)
- Thus, \( f(-x) = 2(1 - 3x + 3x^2 - x^3) = 2 - 6x + 6x^2 - 2x^3 \)
- Since \( f(x) = 2(x + 1)^3 = 2(x^3 + 3x^2 + 3x + 1) = 2x^3 + 6x^2 + 6x + 2 \), \( f(-x) \neq -f(x) \).
- This function is not odd.

3. **\( f(x) = 3x^2 \)**
- \( f(-x) = 3(-x)^2 = 3x^2 = f(x) \)
- This function is even.

4. **\( f(x) = x^4 \)**
- \( f(-x) = (-x)^4 = x^4 = f(x) \)
- This function is also even.

Conclusively, the only odd function in the list is:

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

The function that is not odd is **2. \( f(x) = 2(x + 1)^3 \)**.

Answer 3

Let's carefully re-evaluate the second function to confirm if it is indeed odd or not.

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

To check if it is odd, we need to compute \( f(-x) \) and see if it equals \(-f(x)\):

1. Calculate \( f(-x) \):
\[
f(-x) = 2(-x + 1)^3 = 2(1 - x)^3
\]

Expanding \( (1 - x)^3 \):
\[
(1 - x)^3 = 1 - 3x + 3x^2 - x^3
\]
Therefore,
\[
f(-x) = 2(1 - 3x + 3x^2 - x^3) = 2 - 6x + 6x^2 - 2x^3
\]

2. Now let's compute \( -f(x) \):
\[
f(x) = 2(x + 1)^3 = 2(x^3 + 3x^2 + 3x + 1) = 2x^3 + 6x^2 + 6x + 2
\]
So,
\[
-f(x) = - (2x^3 + 6x^2 + 6x + 2) = -2x^3 - 6x^2 - 6x - 2
\]

3. Now we compare \( f(-x) \) and \(-f(x)\):

- \( f(-x) = 2 - 6x + 6x^2 - 2x^3 \)
- \( -f(x) = -2x^3 - 6x^2 - 6x - 2 \)

Clearly,
\[
f(-x) \neq -f(x)
\]

Thus, \( f(x) = 2(x + 1)^3 \) is neither odd nor even because neither condition holds.

### Summary of Function Parity:
- \( f(x) = 2x^3 \) is odd.
- \( f(x) = 2(x + 1)^3 \) is neither odd nor even.
- \( f(x) = 3x^2 \) is even.
- \( f(x) = x^4 \) is even.

The odd function in your initial list is **1. \( f(x) = 2x^3 \)**. Thank you for your patience as we clarified this point!