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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].


Sagot :

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.