Question

f(x) = -8x⁵ + 4x³ + 5x
a. function f is both even and odd.
b. function f is neither even nor odd.
c. function f is even.
d. function f is odd.

Explanation:

Step1: Recall the definitions of even and odd functions

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

Step2: Compute \( f(-x) \) for the given function \( f(x) = -8x^5 + 4x^3 + 5x \)

Substitute \( -x \) into the function:
\[

$$\begin{align*} f(-x) &= -8(-x)^5 + 4(-x)^3 + 5(-x)\\ &= -8(-x^5) + 4(-x^3) - 5x\\ &= 8x^5 - 4x^3 - 5x \end{align*}$$

\]

Step3: Compare \( f(-x) \) with \( -f(x) \)

First, find \( -f(x) \):
\[
-f(x) = -(-8x^5 + 4x^3 + 5x) = 8x^5 - 4x^3 - 5x
\]
We see that \( f(-x) = -f(x) \).

Answer:

D. Function \( f \) is odd.