Question

Which statement best describes how to determine whether f(x) = x3 + 5x + 1 is an even function?

Answer 1

The right answer is: Determine whether $(-x)^3+5(-x)+1$ is equivalent to $x^3+5x+1$

A function is said to be even if its graph is symmetric with respect to the $y-axis$. That is:

$A \ function \ y=f(x) \ is \ \mathbf{even} \ if, \ for \ each \ x \ in \ the \ domain \ of \ f:\\ \\ f(-x)=f(x)$

According to this definition, the statement that best describes if the function:

$f(x)=x^3+5x+1$

is even, is:

Determine whether $(-x)^3+5(-x)+1$ is equivalent to $x^3+5x+1$

By doing this, we have:

$f(-x)=(-x)^3+5(-x)+1 \\ \\ \therefore \ f(-x)=-x^3-5x+1$

As you can see:

$f(x) \neq f(-x)$

Conclusion: The function is not even.

Answer 2

Its B -Determine whether (–x)3 + 5(–x) + 1 is equivalent to x3 + 5x + 1.