Question

Determine whether the function f, of, x, equals, minus, x, cubed, minus, 7f(x)=−x
3
−7 is even, odd or neither

Answer 1

To determine whether the function \( f(x) = -x^3 - 7 \) is even, odd, or neither, we need to evaluate \( f(-x) \) and compare it to \( f(x) \) and \(-f(x)\).

1. **Calculate \( f(-x) \)**:
\[
f(-x) = -(-x)^3 - 7 = -(-x^3) - 7 = x^3 - 7
\]

2. **Now, compare \( f(-x) \) with \( f(x) \)** and \(-f(x)\)**:
- We already have \( f(x) = -x^3 - 7 \).
- Now compute \(-f(x)\):
\[
-f(x) = -(-x^3 - 7) = x^3 + 7
\]

Now, we have:
- \( f(x) = -x^3 - 7 \)
- \( f(-x) = x^3 - 7 \)
- \(-f(x) = x^3 + 7\)

3. **Check if the function is even**:
A function is even if \( f(-x) = f(x) \).
- Here, \( f(-x) = x^3 - 7 \) is not equal to \( f(x) = -x^3 - 7\), so the function is not even.

4. **Check if the function is odd**:
A function is odd if \( f(-x) = -f(x) \).
- Comparing \( f(-x) = x^3 - 7 \) with \(-f(x) = x^3 + 7\):
\[
x^3 - 7 \neq x^3 + 7
\]
So \( f(-x) \) does not equal \(-f(x)\), and thus the function is not odd.

Since \( f(x) \) is neither even nor odd, we conclude that:

**The function \( f(x) = -x^3 - 7 \) is neither even nor odd.**