Question

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Question
f(x)= -x^5 - 6x^3
x y
-2 80
-1 7
0 0
1 -7
2 -80

Select whether the function above is odd, even or neither
(1 point)
Responses

odd
odd

even
even

neither

Answer 1

To determine whether the function \( f(x) = -x^5 - 6x^3 \) is odd, even, or neither, we will use the definitions of odd and even functions:

1. **Even Function:** A function is even if \( f(-x) = f(x) \) for all \( x \).
2. **Odd Function:** A function is odd if \( f(-x) = -f(x) \) for all \( x \).

Let's check the values of the function provided for \( f(-x) \):

1. Calculate \( f(-2) \):
\[
f(-2) = -(-2)^5 - 6(-2)^3 = -(-32) - 6(-8) = 32 + 48 = 80
\]

2. Calculate \( f(-1) \):
\[
f(-1) = -(-1)^5 - 6(-1)^3 = -(-1) - 6(-1) = 1 + 6 = 7
\]

3. Calculate \( f(0) \):
\[
f(0) = -0^5 - 6(0)^3 = 0
\]

4. Calculate \( f(1) \):
\[
f(1) = -(1)^5 - 6(1)^3 = -1 - 6 = -7
\]

5. Calculate \( f(2) \):
\[
f(2) = -(2)^5 - 6(2)^3 = -32 - 48 = -80
\]

Now, we compare \( f(-x) \) with \( -f(x) \):
- For \( x = 2 \):
\[
f(-2) = 80 \quad \text{and} \quad -f(2) = -(-80) = 80
\]
- For \( x = 1 \):
\[
f(-1) = 7 \quad \text{and} \quad -f(1) = -(-7) = 7
\]
- For \( x = 0 \):
\[
f(0) = 0 \quad \text{and} \quad -f(0) = -0 = 0
\]

Since for all values checked \( f(-x) = -f(x) \), the function \( f(x) = -x^5 - 6x^3 \) is confirmed to be an odd function.

Thus, the correct response is:
**odd**