To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is even, we need to check if it satisfies the definition of an even function. A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex].
Let's go through the steps to check this:
1. Substitute [tex]\( -x \)[/tex] into the function [tex]\( f(x) \)[/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]
4. Determine whether [tex]\( f(-x) \)[/tex] is equivalent to [tex]\( f(x) \)[/tex]:
[tex]\[
-x^3 - 5x + 1 \neq x^3 + 5x + 1
\][/tex]
Since [tex]\( f(-x) \)[/tex] is not equal to [tex]\( f(x) \)[/tex], the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is not an even function.
Thus, the best statement that describes how to determine whether [tex]\( f(x) = x^3 + 5 x + 1 \)[/tex] is an even function is:
- Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5 x + 1 \)[/tex].
So, the correct option is:
- Determine whether [tex]\((-x)^3 + 5(-x) + 1\)[/tex] is equivalent to [tex]\(x^3 + 5 x + 1\)[/tex].
