Answer:
C.
Step-by-step explanation:
We are given [tex]f,g[/tex] are odd which means:
[tex]f(-x)=-f(x)[/tex]
[tex]g(-x)=-g(x)[/tex]
We can tell if a function,[tex]h[/tex], is even if [tex]h(-x)=h(x)[/tex].
We can tell if a function,[tex]h[/tex], is odd if [tex]h(-x)=-h(x)[/tex].
So let's test your I,II,III.
We will be replacing x with -x to find out.
I.
[tex]p(x)=f(g(x))[/tex]
[tex]p(-x)=f(g(-x))[/tex]
[tex]p(-x)=f(-g(x))[/tex]
[tex]p(-x)=-f(g(x))[/tex]
[tex]p(-x)=-p(x)[/tex]
So [tex]p[/tex] is odd.
II
[tex]r(x)=f(x)+g(x)[/tex]
[tex]r(-x)=f(-x)+g(-x)[/tex]
[tex]r(-x)=-f(x)+-g(x)[/tex]
[tex]r(-x)=-(f(x)+g(x))[/tex]
[tex]r(-x)=-(r(x))[/tex]
[tex]r(-x)=-r(x)[/tex]
So [tex]r[/tex] is odd.
III
[tex]s(x)=f(x)\cdot g(x)[/tex]
[tex]s(-x)=f(-x) \cdot g(-x)[/tex]
[tex]s(-x)=-f(x) \cdot -g(x)[/tex]
[tex]s(-x)=f(x) \cdot g(x)[/tex]
[tex]s(-x)=s(x)[/tex]
So [tex]s[/tex] is even.
So I and II are odd and III is even.
C. is the answer.
