Which statement best describes how to determine whether [tex]\( f(x) = x^2 - x + 8 \)[/tex] is an even function?

A. Determine whether [tex]\( -x^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( x^2 - x + 8 \)[/tex].

B. Determine whether [tex]\( (-x)^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( x^2 - x + 8 \)[/tex].

C. Determine whether [tex]\( -x^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( -\left(x^2 - x + 8\right) \)[/tex].

D. Determine whether [tex]\( (-x)^2 - (-x) + 8 \)[/tex] is equivalent to [tex]\( -\left(x^2 - x + 8\right) \)[/tex].



Answer :

To determine if the function [tex]\( f(x) = x^2 - x + 8 \)[/tex] is an even function, you need to evaluate [tex]\( f(-x) \)[/tex] and compare it to [tex]\( f(x) \)[/tex].

1. Start by substituting [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function:
[tex]\[ f(-x) = (-x)^2 - (-x) + 8. \][/tex]

2. Simplify the expression:
[tex]\[ (-x)^2 - (-x) + 8 = x^2 + x + 8. \][/tex]

3. Now, compare [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^2 - x + 8. \][/tex]
[tex]\[ f(-x) = x^2 + x + 8. \][/tex]

Clearly, [tex]\( f(-x) \neq f(x) \)[/tex].

Therefore, the function [tex]\( f(x) = x^2 - x + 8 \)[/tex] is not an even function.

From the given statements, the one that correctly describes the method is:
[tex]\[ \text{Determine whether } (-x)^2 - (-x) + 8 \text{ is equivalent to } x^2 - x + 8. \][/tex]

Thus, the correct statement is:
[tex]\[ \text{Determine whether } (-x)^2 - (-x) + 8 \text{ is equivalent to } x^2 - x + 8. \][/tex]