Question

is the function n(x) = 2x⁵ + x³ even, odd, or neither?
even
odd
neither

Explanation:

Step1: Recall the definitions

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

Step2: Compute \( n(-x) \)

Given \( n(x) = 2x^5 + x^3 \), substitute \( -x \) for \( x \):
\( n(-x) = 2(-x)^5 + (-x)^3 \)

Step3: Simplify \( n(-x) \)

Using the property \( (-a)^k = -a^k \) when \( k \) is odd:
\( (-x)^5 = -x^5 \) and \( (-x)^3 = -x^3 \). So,
\( n(-x) = 2(-x^5) + (-x^3) = -2x^5 - x^3 \)

Step4: Compare with \( -n(x) \)

Compute \( -n(x) \):
\( -n(x) = -(2x^5 + x^3) = -2x^5 - x^3 \)

Since \( n(-x) = -n(x) \), the function is odd.

Answer:

odd