To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is an even function, we need to verify if [tex]\( f(-x) = f(x) \)[/tex] holds true for all [tex]\( x \)[/tex].
To proceed:
1. Calculate [tex]\( f(-x) \)[/tex] from the given function:
[tex]\[
f(x) = x^3 + 5x + 1
\][/tex]
Substituting [tex]\( -x \)[/tex] in place of [tex]\( x \)[/tex]:
[tex]\[
f(-x) = (-x)^3 + 5(-x) + 1
\][/tex]
2. Simplify [tex]\( f(-x) \)[/tex]:
[tex]\[
(-x)^3 + 5(-x) + 1 = -x^3 - 5x + 1
\][/tex]
3. Compare [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^3 + 5x + 1
\][/tex]
[tex]\[
f(-x) = -x^3 - 5x + 1
\][/tex]
For [tex]\( f(x) \)[/tex] to be an even function, [tex]\( f(-x) \)[/tex] should equal [tex]\( f(x) \)[/tex]:
[tex]\[
-x^3 - 5x + 1 \stackrel{?}{=} x^3 + 5x + 1
\][/tex]
It is evident that [tex]\( f(-x) \)[/tex] does not equal [tex]\( f(x) \)[/tex], as [tex]\(-x^3 - 5x + 1 \neq x^3 + 5x + 1\)[/tex].
Therefore, the correct statement to determine whether [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is an even function is:
[tex]\[
\text{Determine whether } (-x)^3 + 5(-x) + 1 \text{ is equivalent to } x^3 + 5x + 1.
\][/tex]
Thus, the best description to check if [tex]\( f(x) \)[/tex] is an even function involves checking the equivalence of [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex], which corresponds to:
Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5x + 1 \)[/tex].
So, the correct answer is:
[tex]\[
2 \quad \text{(Determine whether } (-x)^3 + 5(-x) + 1 \text{ is equivalent to } x^3 + 5x + 1\text{)}
\][/tex]
