Which statement describes the function f left-parenthesis x right-parenthesis equals Start Absolute Value x superscript 4 baseline minus x superscript 3 baseline End Absolute Value?

Question

(1 point)
Responses

The function is neither even nor odd.
The function is neither even nor odd.

The function is both even and odd.
The function is both even and odd.

The function is even.
The function is even.

The function is odd.

Answer 1

To determine whether the function \( f(x) = |x^4 - x^3| \) is even, odd, or neither, we need to evaluate \( f(-x) \) and compare it with \( f(x) \).

1. First, calculate \( f(-x) \): \[ f(-x) = |-x^4 - (-x^3)| = |-x^4 + x^3| = |-(x^4 - x^3)| = |x^4 - x^3| = f(x) \]

Since \( f(-x) = f(x) \), this shows that the function is even.

2. Next, consider if it could be odd. A function is odd if \( f(-x) = -f(x) \). Since we found \( f(-x) = f(x) \), it cannot be odd.

Based on this analysis, the correct statement is:

The function is even.