The right answer is: Determine whether [tex](-x)^3+5(-x)+1[/tex] is equivalent to [tex]x^3+5x+1[/tex]
A function is said to be even if its graph is symmetric with respect to the [tex]y-axis[/tex]. That is:
[tex]A \ function \ y=f(x) \ is \ \mathbf{even} \ if, \ for \ each \ x \ in \ the \ domain \ of \ Â f:\\ \\ f(-x)=f(x)[/tex]
According to this definition, the statement that best describes if the function:
[tex]f(x)=x^3+5x+1[/tex]
is even, is:
Determine whether [tex](-x)^3+5(-x)+1[/tex] is equivalent to [tex]x^3+5x+1[/tex]
By doing this, we have:
[tex]f(-x)=(-x)^3+5(-x)+1 \\ \\ \therefore \ f(-x)=-x^3-5x+1[/tex]
As you can see:
[tex]f(x) \neq f(-x)[/tex]
Conclusion: The function is not even.
