To determine whether the function [tex]\( f(x) = x^4 - x^3 \)[/tex] is an even function, you need to examine whether [tex]\( f(-x) = f(x) \)[/tex].
Let's explore each statement step by step:
1. Determine whether [tex]\( (-x)^4 - (-x)^3 \)[/tex] is equivalent to [tex]\( x^4 - x^3 \)[/tex]:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^4 - (-x)^3 = x^4 + x^3
\][/tex]
- Check if [tex]\( f(-x) = f(x) \)[/tex]:
[tex]\[
x^4 + x^3 \neq x^4 - x^3
\][/tex]
- This statement suggests [tex]\( f(-x) \)[/tex] is equivalent to [tex]\( f(x) \)[/tex]. The calculation shows that this is false.
2. Determine whether [tex]\( \left(-x^4\right) - \left(-x^3\right) \)[/tex] is equivalent to [tex]\( x^4 + x^3 \)[/tex]:
- This transforms into:
[tex]\[
-x^4 + x^3
\][/tex]
- Check if this is equivalent to [tex]\( x^4 + x^3 \)[/tex]:
[tex]\[
-x^4 + x^3 \neq x^4 + x^3
\][/tex]
- Thus, this statement is false.
3. Determine whether [tex]\( (-x)^4 - (-x)^3 \)[/tex] is equivalent to [tex]\(-\left(x^4 - x^3\right)\)[/tex]:
- Calculate:
[tex]\[
(-x)^4 - (-x)^3 = x^4 + x^3
\][/tex]
- Check if this is equivalent to:
[tex]\[
-\left(x^4 - x^3\right) = -x^4 + x^3
\][/tex]
- The expression [tex]\( x^4 + x^3 \neq -x^4 + x^3 \)[/tex]
- Hence, this statement is also false.
4. Determine whether [tex]\(\left( - x^4 \right) - \left( - x^3 \right)\)[/tex] is equivalent to [tex]\(-\left(x^4 + x^3\right)\)[/tex]:
- This transforms into:
[tex]\[
-x^4 + x^3
\][/tex]
- Compare it with:
[tex]\[
-\left(x^4 + x^3\right) = -x^4 - x^3
\][/tex]
- So, [tex]\(-x^4 + x^3 \neq -x^4 - x^3 \)[/tex]
- Again, this statement is false.
Given these calculations and comparisons, we can conclude that all the four provided statements are false. The correct approach to determine whether [tex]\( f(x) \)[/tex] is an even function is to check if [tex]\( f(-x) = f(x) \)[/tex], and in this case, it is not true. Thus, the function [tex]\( f(x) = x^4 - x^3 \)[/tex] is not an even function.
