Question

I don't know what to do. This is about odd and even functions.

Answer 1

The answer will be C

Answer 2

C.

Explanation:

We are given
`f,g` are odd which means:

`f(-x)=-f(x)`

`g(-x)=-g(x)`

We can tell if a function,
`h`, is even if
`h(-x)=h(x)`.

We can tell if a function,
`h`, is odd if
`h(-x)=-h(x)`.

So let's test your I,II,III.

We will be replacing x with -x to find out.

I.

`p(x)=f(g(x))`

`p(-x)=f(g(-x))`

`p(-x)=f(-g(x))`

`p(-x)=-f(g(x))`

`p(-x)=-p(x)`

So
`p` is odd.

II

`r(x)=f(x)+g(x)`

`r(-x)=f(-x)+g(-x)`

`r(-x)=-f(x)+-g(x)`

`r(-x)=-(f(x)+g(x))`

`r(-x)=-(r(x))`

`r(-x)=-r(x)`

So
`r` is odd.

III

`s(x)=f(x)\cdot g(x)`

`s(-x)=f(-x) \cdot g(-x)`

`s(-x)=-f(x) \cdot -g(x)`

`s(-x)=f(x) \cdot g(x)`

`s(-x)=s(x)`

So
`s` is even.

So I and II are odd and III is even.

C. is the answer.