Question

Determine whether the function f, of, x, equals, x, to the power 5 , minus, x, cubed, minus, 6, xf(x)=x
5
−x
3
−6x is even, odd or neither.
Answer
Attempt 1 out of 2
Multiple Choice Answers
odd
even
neither

Answer 1

To determine if the function \( f(x) = x^5 - x^3 - 6x \) is even, odd, or neither, we need to check the following:

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

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

\[
f(-x) = (-x)^5 - (-x)^3 - 6(-x)
\]
\[
= -x^5 + x^3 + 6x
\]

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

We already have
\[
f(x) = x^5 - x^3 - 6x
\]

Now, let's find \( -f(x) \):
\[
-f(x) = -(x^5 - x^3 - 6x) = -x^5 + x^3 + 6x
\]

Now we can compare:
- We see that \( f(-x) = -f(x) \).

Since \( f(-x) = -f(x) \), this means that the function \( f(x) \) is **odd**.

The answer is:
**odd**.