The alternatives indicates that the question is aboud even and odd function.
An even function is one the gives:
[tex]f(x)=f(-x)[/tex]
While an odd function is one tha gives:
[tex]f(x)=-f(-x)[/tex]
The fisrt function is:
[tex]\begin{gathered} f(x)=\cos ^3(x^3-x) \\ f(-x)=\cos ^3((-x)^3-(-x))=\cos ^3(-x^3+x)=\cos ^{}^{3}(-(x^3-x)) \end{gathered}[/tex]
Since
[tex]\cos (x)=\cos (-x)[/tex]
Then
[tex]\begin{gathered} \cos ^3(-(x^3-x))=\cos ^3(x^3-x) \\ f(x)=f(-x) \end{gathered}[/tex]
So, f is even.
We can do it similarly for g(x)
[tex]\begin{gathered} g(x)=\ln (|x|+3) \\ g(-x)=\ln (|-x|+3)=\ln (|x|+3) \\ g(x)=g(-x) \end{gathered}[/tex]
For s(x), we have:
[tex]\begin{gathered} s(x)=\sin ^{3}(x) \\ s(-x)=\sin ^{3}(-x) \end{gathered}[/tex]
Since:
[tex]\sin (-x)=-\sin (x)[/tex]
Then:
[tex]\begin{gathered} \sin ^3(-x)=(-\sin (x))^3=-\sin ^{3}(x) \\ s(x)=-s(-x) \end{gathered}[/tex]
So far we have f(x) even, g(x) even and s(x) odd. This is exactly what is said in alternative A:
A. f and g are even, s is odd.
so that is the right answer.
