Question

Question
(1 point)
Responses
determine whether the function f(x) = -x^3 - x -3 is even, odd or neither
even
even

odd
odd

neither

Answer 1

To determine whether the function \( f(x) = -x^3 - x - 3 \) is even, odd, or neither, we can apply the definitions of even and odd functions:

- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

Let's compute \( f(-x) \):

\[
f(-x) = -(-x)^3 - (-x) - 3
\]
\[
= -(-x^3) + x - 3
\]
\[
= x^3 + x - 3
\]

Now, we compare \( f(-x) \) with \( -f(x) \):

First, calculate \( -f(x) \):

\[
-f(x) = -(-x^3 - x - 3) = x^3 + x + 3
\]

Now we have:
- \( f(-x) = x^3 + x - 3 \)
- \( -f(x) = x^3 + x + 3 \)

Since \( f(-x) \) is not equal to \( f(x) \) (which would indicate that it's even) and \( f(-x) \) is not equal to \( -f(x) \) (which would indicate that it's odd), we conclude that:

The function \( f(x) = -x^3 - x - 3 \) is **neither** even nor odd.