Which statement best describes how to determine whether [tex]f(x) = x^3 + 5x + 1[/tex] is an even function?

A. Determine whether [tex]-\left(x^3 + 5x + 1\right)[/tex] is equivalent to [tex]x^3 + 5x + 1[/tex].
B. Determine whether [tex](-x)^3 + 5(-x) + 1[/tex] is equivalent to [tex]x^3 + 5x + 1[/tex].
C. Determine whether [tex]-x^3 + 5x + 1[/tex] is equivalent to [tex]-\left(x^3 + 5x + 1\right)[/tex].
D. Determine whether [tex](-x)^3 + 5(-x) + 1[/tex] is equivalent to [tex]-\left(x^3 + 5x + 1\right)[/tex].



Answer :

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]