Question

which statement best describes how to determine whether $f(x)=x^3 + 5x + 1$ is an even function?
determine whether $(-x)^3 + 5(-x) + 1$ is equivalent to $x^3 + 5x + 1$.
determine whether $-x^3 + 5x + 1$ is equivalent to $-(x^3 + 5x + 1)$.
determine whether $(-x)^3 + 5(-x) + 1$ is equivalent to $-(x^3 + 5x + 1)$.
determine whether $-(x^3 + 5x + 1)$ is equivalent to $x^3 + 5x + 1$.

Explanation:

To determine if a function \( f(x) \) is even, we use the definition of an even function: \( f(-x)=f(x) \) for all \( x \) in the domain. For the function \( f(x)=x^{3}+5x + 1 \), we need to find \( f(-x) \) by substituting \( -x \) for \( x \) in the function, which gives \( f(-x)=(-x)^{3}+5(-x)+1 \). Then we check if this \( f(-x) \) is equivalent to \( f(x)=x^{3}+5x + 1 \).

• The second option checks for \( -x^{3}+5x + 1=- (x^{3}+5x + 1) \), which is the condition for an odd function (\( f(-x)=-f(x) \)), not even.
• The third option also checks against \( - (x^{3}+5x + 1) \), which is for odd functions.
• The fourth option checks \( - (x^{3}+5x + 1) \) against \( x^{3}+5x + 1 \), which is not related to the even function definition.

Only the first option correctly follows the even - function definition by checking if \( f(-x)=(-x)^{3}+5(-x)+1 \) is equivalent to \( f(x)=x^{3}+5x + 1 \).

Answer:

Determine whether \( (-x)^{3}+5(-x)+1 \) is equivalent to \( x^{3}+5x + 1 \). (The first option among the given boxes)