To determine whether [tex]\( f(x) = x^2 - x + 8 \)[/tex] is an even function, we need to check whether [tex]\( f(-x) \)[/tex] is equivalent to [tex]\( f(x) \)[/tex].
Let's take the candidate statement "Determine whether [tex]\( (-x)^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( x^2 - x + 8 \)[/tex]":
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^2 - (-x) + 8
\][/tex]
2. Simplify [tex]\( f(-x) \)[/tex]:
[tex]\[
(-x)^2 = x^2 \quad \text{(since squaring a number results in a positive value)}
\][/tex]
[tex]\[
-(-x) = x \quad \text{(double negative results in a positive)}
\][/tex]
So,
[tex]\[
f(-x) = x^2 + x + 8
\][/tex]
3. Compare [tex]\( f(-x) \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^2 - x + 8
\][/tex]
[tex]\[
f(-x) = x^2 + x + 8
\][/tex]
By comparing [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex], we see they are not equivalent, because [tex]\( x^2 - x + 8 \)[/tex] is not equal to [tex]\( x^2 + x + 8 \)[/tex].
Hence, the correct approach statement is:
Determine whether [tex]\( (-x)^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( x^2 - x + 8 \)[/tex].
This statement is correct because it directly leads to comparing [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex] to see if the function is even. Since [tex]\( f(-x) \)[/tex] is not equivalent to [tex]\( f(x) \)[/tex], [tex]\( f(x) \)[/tex] is not an even function.
