To determine whether the function [tex]\( f(x) = x^4 - x^3 \)[/tex] is an even function, we need to verify if [tex]\( f(-x) = f(x) \)[/tex]. Let's proceed step-by-step:
1. Substitute [tex]\( -x \)[/tex] into the function:
[tex]\[
f(-x) = (-x)^4 - (-x)^3
\][/tex]
2. Simplify the expression:
[tex]\[
(-x)^4 = x^4 \quad \text{because} \quad (-x)^4 = (-x)(-x)(-x)(-x) = x^4
\][/tex]
[tex]\[
(-x)^3 = -x^3 \quad \text{because} \quad (-x)^3 = (-x)(-x)(-x) = -x^3
\][/tex]
[tex]\[
f(-x) = x^4 - (-x^3) = x^4 + x^3
\][/tex]
3. Compare [tex]\( f(-x) \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^4 - x^3
\][/tex]
[tex]\[
f(-x) = x^4 + x^3
\][/tex]
By comparing the original function [tex]\( f(x) = x^4 - x^3 \)[/tex] with [tex]\( f(-x) = x^4 + x^3 \)[/tex], we see that [tex]\( f(-x) \)[/tex] is not identical to [tex]\( f(x) \)[/tex].
Therefore, [tex]\( f(x) = x^4 - x^3 \)[/tex] is not an even function.
Considering the provided statements, the correct one that describes the process to determine whether [tex]\( f(x) = x^4 - x^3 \)[/tex] is an even function is:
Determine whether [tex]\( (-x)^4 - (-x)^3 \)[/tex] is equivalent to [tex]\( x^4 - x^3 \)[/tex].
Based on our calculations, the answer is False, so the function is not even.
