Is the function f(x)=(x−2)3+1

Question

an odd function?

(1 point)
Responses

No, because the graphs of f(−x)
and −f(x)
are not the same
No, because the graphs of f of negative x and negative f times x are not the same

Yes, because f(x)
has both positive and negative values
Yes, because f of x has both positive and negative values

No, because the function contains even numbers
No, because the function contains even numbers

Yes, because the highest exponent is 3, which is an odd number

Answer 1

To determine if the function \( f(x) = (x - 2)^3 + 1 \) is an odd function, we need to check if it satisfies the condition for odd functions, which is:

\[ f(-x) = -f(x) \]

1. Calculate \( f(-x) \): \[ f(-x) = (-x - 2)^3 + 1 \] Simplifying \( (-x - 2)^3 \): \[ -x - 2 = -(x + 2) \Rightarrow (-x - 2)^3 = -(x + 2)^3 \] Expanding \( (x + 2)^3 \): \[ (x + 2)^3 = x^3 + 3(2)x^2 + 3(2^2)x + 2^3 = x^3 + 6x^2 + 12x + 8 \] Thus, \[ (-x - 2)^3 = -(x^3 + 6x^2 + 12x + 8) = -x^3 - 6x^2 - 12x - 8 \]

   Therefore, \[ f(-x) = -x^3 - 6x^2 - 12x - 8 + 1 = -x^3 - 6x^2 - 12x - 7 \]

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

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

From the calculations:

• \( f(-x) = -x^3 - 6x^2 - 12x - 7 \)

• \( -f(x) = -((x - 2)^3 + 1) = -(x^3 - 6x^2 + 12x - 8) - 1 = -x^3 + 6x^2 - 12x + 7 \)

We can see that \( f(-x) \) is not equal to \( -f(x) \).

Therefore, the correct response is:

No, because the graphs of \( f(-x) \) and \( -f(x) \) are not the same.