Question

which statement best describes how to determine whether $f(x) = x^2 - x + 8$ is an even function?

determine whether $(-x)^2 - (-x) + 8$ is equivalent to $-(x^2 - x + 8)$.

determine whether $(-x)^2 - (-x) + 8$ is equivalent to $x^2 - x + 8$.

determine whether $-x^2 - (-x) + 8$ is equivalent to $-(x^2 - x + 8)$.

determine whether $-x^2 - (-x) + 8$ is equivalent to $x^2 - x + 8$.

Explanation:

To determine if a function \( f(x) \) is even, we use the definition of an even function: \( f(-x) = f(x) \).

Step 1: Recall the definition of an even function

A function \( f(x) \) is even if for all \( x \) in the domain, \( f(-x) = f(x) \). So we need to find \( f(-x) \) and check if it equals \( f(x) \).

Step 2: Find \( f(-x) \) for \( f(x) = x^2 - x + 8 \)

Substitute \( -x \) into the function:
\( f(-x) = (-x)^2 - (-x) + 8 \)

Step 3: Check the condition for an even function

We need to determine if \( f(-x) \) (which is \( (-x)^2 - (-x) + 8 \)) is equivalent to \( f(x) \) (which is \( x^2 - x + 8 \)).

Now let's analyze the options:

• Option 1: Checks if \( f(-x) = -f(x) \), which is for odd functions, not even.
• Option 2: Checks if \( f(-x) = f(x) \), which is the definition of an even function.
• Option 3: Incorrectly computes \( f(-x) \) (wrong sign on \( x^2 \)) and checks against \( -f(x) \), not relevant.
• Option 4: Incorrectly computes \( f(-x) \) (wrong sign on \( x^2 \)) and checks against \( f(x) \), but the computation of \( f(-x) \) is wrong.

Answer:

The correct option is the second one: "Determine whether \((-x)^2 - (-x) + 8\) is equivalent to \(x^2 - x + 8\)."