To determine whether [tex]\( f(x) = x^4 - x^3 \)[/tex] is an even function, we start by evaluating [tex]\( f(-x) \)[/tex]:
First, substitute [tex]\(-x\)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-x) = (-x)^4 - (-x)^3 \][/tex]
Next, simplify the expressions inside the function:
[tex]\[ (-x)^4 = (-x) \cdot (-x) \cdot (-x) \cdot (-x) = x^4 \][/tex]
[tex]\[ (-x)^3 = (-x) \cdot (-x) \cdot (-x) = -x^3 \][/tex]
Now substitute these simplified expressions back into our function:
[tex]\[ f(-x) = x^4 - (-x^3) = x^4 + x^3 \][/tex]
We need to compare [tex]\( f(-x) = x^4 + x^3 \)[/tex] with the original function [tex]\( f(x) = x^4 - x^3 \)[/tex].
Since [tex]\( f(-x) = x^4 + x^3 \)[/tex] is not equivalent to [tex]\( f(x) = x^4 - x^3 \)[/tex], the function [tex]\( f(x) \)[/tex] is not even.
Therefore, the correct statement to determine whether [tex]\( f(x) = x^4 - x^3 \)[/tex] is an even function is:
[tex]\[ \text{Determine whether } (-x)^4 - (-x)^3 \text{ is equivalent to } x^4 - x^3. \][/tex]
This statement corresponds to the option:
[tex]\[ \text{Determine whether } (-x)^4 - (-x)^3 \text{ is equivalent to } x^4 - x^3. \][/tex]
The result of comparing [tex]\( (-x)^4 - (-x)^3 \)[/tex] with [tex]\( x^4 - x^3 \)[/tex] is that they are not equivalent, hence the function is not even.
