Question

is the function $f(x) = x^4 + x$ even?
use the drop-down menus to complete the statements.
click the arrows to choose an answer from each menu.
to determine if $f(x)$ is even, find $f(-x) =$ choose... . because $f(-x)$ choose... equal to $f(x)$, the function choose... even.

Explanation:

Step1: Find \( f(-x) \)

Given \( f(x) = x^4 + x \), substitute \( x \) with \( -x \):
\( f(-x) = (-x)^4 + (-x) \)
Simplify: \( (-x)^4 = x^4 \) (since even power), and \( -x \) remains. So \( f(-x) = x^4 - x \).

Step2: Compare \( f(-x) \) and \( f(x) \)

\( f(x) = x^4 + x \), and \( f(-x) = x^4 - x \). These are not equal (e.g., test \( x = 1 \): \( f(1) = 2 \), \( f(-1) = 0 \)).

Step3: Determine if even

A function is even if \( f(-x) = f(x) \) for all \( x \). Since \( f(-x)
eq f(x) \), the function is not even.

Answer:

To determine if \( f(x) \) is even, find \( f(-x) = \boldsymbol{x^4 - x} \). Because \( f(-x) \) is \(\boldsymbol{\text{not}}\) equal to \( f(x) \), the function is \(\boldsymbol{\text{not}}\) even.