To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is an even function, you need to compare the values of [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex].
The correct approach is to determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5x + 1 \)[/tex].
Let's break this down step-by-step:
1. Substitute [tex]\( -x \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = (-x)^3 + 5(-x) + 1
\][/tex]
2. Simplify [tex]\( f(-x) \)[/tex]:
[tex]\[
(-x)^3 = -x^3
\][/tex]
[tex]\[
5(-x) = -5x
\][/tex]
Substituting these into the equation gives:
[tex]\[
f(-x) = -x^3 - 5x + 1
\][/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]
As you can see, these two expressions are not equivalent. For [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] to be an even function, [tex]\( f(-x) \)[/tex] must equal [tex]\( f(x) \)[/tex]. Since [tex]\( -x^3 - 5x + 1 \)[/tex] is not the same as [tex]\( x^3 + 5x + 1 \)[/tex], the function [tex]\( f(x) \)[/tex] is not an even function.
Hence, the correct statement is:
"Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5 x + 1 \)[/tex]."
