To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is even, you need to compare [tex]\( f(x) \)[/tex] with [tex]\( f(-x) \)[/tex].
A function [tex]\( f(x) \)[/tex] is even if:
[tex]\[ f(x) = f(-x) \][/tex]
Starting with [tex]\( f(-x) \)[/tex], let's substitute [tex]\(-x\)[/tex] into the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] and simplify:
[tex]\[ f(-x) = (-x)^3 + 5(-x) + 1 \][/tex]
Simplifying [tex]\( f(-x) \)[/tex]:
[tex]\[ (-x)^3 = -x^3 \][/tex]
[tex]\[ 5(-x) = -5x \][/tex]
So:
[tex]\[ f(-x) = -x^3 - 5x + 1 \][/tex]
Now we 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]
Clearly, [tex]\( f(-x) \)[/tex] is not equivalent to [tex]\( f(x) \)[/tex] because:
[tex]\[ x^3 + 5x + 1 \neq -x^3 - 5x + 1 \][/tex]
This means that [tex]\( f(x) \)[/tex] is not an even function.
Therefore, the statement that best describes how to determine whether [tex]\( f(x) \)[/tex] is even is:
Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5x + 1 \)[/tex].
Since [tex]\( f(-x) \neq f(x) \)[/tex], the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is not even.
