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

ADetermine whether $(-x)^2-(-x)+8$ is equivalent to $-\left(x^2-x+8\right)$.
BDetermine whether $-x^2-(-x)+8$ is equivalent to $-\left(x^2-x+8\right)$.
CDetermine whether $-x^2-(-x)+8$ is equivalent to $x^2-x+8$.
DDetermine whether $(-x)^2-(-x)+8$ is equivalent to $x^2-x+8$.

Question

03-11-2026
Mathematics

Answered

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

ADetermine whether $(-x)^2-(-x)+8$ is equivalent to $-\left(x^2-x+8\right)$.
BDetermine whether $-x^2-(-x)+8$ is equivalent to $-\left(x^2-x+8\right)$.
CDetermine whether $-x^2-(-x)+8$ is equivalent to $x^2-x+8$.
DDetermine whether $(-x)^2-(-x)+8$ is equivalent to $x^2-x+8$.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T18:09:08-05:00

The definition of an even function is $f(-x) = f(x)$.
Substitute $-x$ into the given function $f(x) = x^2 - x + 8$ to get $f(-x) = (-x)^2 - (-x) + 8$.
Compare the resulting expression $f(-x) = (-x)^2 - (-x) + 8$ with the original $f(x) = x^2 - x + 8$.
The correct statement is: $\boxed{\text{Determine whether } (-x)^2 - (-x) + 8 \text{ is equivalent to } x^2 - x + 8}$.

Explanation

Defining the Condition for an Even Function

To determine whether a function $f(x)$ is an even function, we must test if the condition $f(-x) = f(x)$ holds for all values of $x$ in its domain. The given function is $f(x) = x^2 - x + 8$.

Evaluating the Expression and Comparing

To test the condition, we need to evaluate the function at $-x$. We take the expression $f(x) = x^2 - x + 8$ and substitute $-x$ for every occurrence of $x$. This gives us:
\[f(-x) = (-x)^2 - (-x) + 8\]
Next, we identify the expression for the original function, which is $f(x) = x^2 - x + 8$. To check if the function is even, we compare these two expressions by checking if they are equivalent, meaning we need to determine if:
\[(-x)^2 - (-x) + 8 = x^2 - x + 8\]

Conclusion

Reviewing the provided choices:

Determine whether $(-x)^2 - (-x) + 8$ is equivalent to $-\left(x^2 - x + 8\right)$.
Determine whether $-x^2 - (-x) + 8$ is equivalent to $-\left(x^2 - x + 8\right)$.
Determine whether $-x^2 - (-x) + 8$ is equivalent to $x^2 - x + 8$.
Determine whether $(-x)^2 - (-x) + 8$ is equivalent to $x^2 - x + 8$.

The correct procedure is to compare $f(-x)$ with $f(x)$. Substituting $-x$ into the function leads to $(-x)^2 - (-x) + 8$, and comparing this to the original function $x^2 - x + 8$ confirms that the final statement provided is the correct one. Therefore, the statement "Determine whether $(-x)^2 - (-x) + 8$ is equivalent to $x^2 - x + 8$" is the correct description.

Examples

In many fields like engineering or physics, identifying properties of functions like symmetry (even or odd) can simplify complex calculations. For example, when calculating the area under a curve or solving differential equations, recognizing that a function is even (symmetric about the y-axis) allows you to integrate over only half the domain and multiply by two. By understanding the formal test $f(-x) = f(x)$, one can quickly identify these symmetric behaviors in various physical models, such as the parabolic path of a projectile or the oscillations of a pendulum.