Question

Determine whether the function f, of, x, equals, x, squared, minus, 8, x, to the power 6f(x)=x
2
−8x
6
is even, odd or neither.
Answer
Attempt 1 out of 2
Multiple Choice Answers
odd
even
neither

Answer 1

To determine whether the function \( f(x) = x^2 - 8x^6 \) is even, odd, or neither, we can use the definitions of even and odd functions:

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

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

\[
f(-x) = (-x)^2 - 8(-x)^6
\]

Calculating each term:

\[
(-x)^2 = x^2
\]
\[
(-x)^6 = x^6 \quad \text{(since the exponent is even)}
\]

Now substitute these back into \( f(-x) \):

\[
f(-x) = x^2 - 8x^6
\]

Now we compare \( f(-x) \) to \( f(x) \):

\[
f(-x) = x^2 - 8x^6 \quad \text{and}\quad f(x) = x^2 - 8x^6
\]

Since \( f(-x) = f(x) \), we conclude that the function \( f(x) \) is **even**.

**Final Answer: even**