Ask a New Question
Menu

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

3 answers

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

2. be correct

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

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

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!