Question

1. determine whether the function below is even, odd, or neither. f(m) = 4x⁵ - 5x⁴ a) cannot be determined b) odd c) even d) neither

Explanation:

Step1: Recall definitions

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

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

Given \( f(x) = 4x^5 - 5x^4 \) (note: the variable is \( x \), not \( m \) as written, likely a typo), substitute \( -x \) for \( x \):
\( f(-x) = 4(-x)^5 - 5(-x)^4 \)
Simplify each term:
\( (-x)^5 = -x^5 \), so \( 4(-x)^5 = 4(-x^5) = -4x^5 \)
\( (-x)^4 = x^4 \), so \( -5(-x)^4 = -5(x^4) = -5x^4 \)
Thus, \( f(-x) = -4x^5 - 5x^4 \)

Step3: Compare to \( f(x) \) and \( -f(x) \)

• \( f(x) = 4x^5 - 5x^4 \). \( f(-x)

eq f(x) \) (since \( -4x^5
eq 4x^5 \)), so not even.

• \( -f(x) = - (4x^5 - 5x^4) = -4x^5 + 5x^4 \). \( f(-x) = -4x^5 - 5x^4

eq -f(x) \) (since \( -5x^4
eq +5x^4 \)), so not odd.

Answer:

d) neither