Question

let f and g are functions that are neither even nor odd. a)Create an example where f+g is even, b) f+g is odd, c)f.g is even, d) f.g is odd

Answer 1

I give you one case , b, as example, you do the rest, ok?
Let \(f(x) = x+3\) let check
\(f(-x) = -x +3 \neq f(x) ~~hence,~~\text{f(x) is not even}\)
\(-f(x) = -x-3 \neq f(-x)~~hence,~~\text{f(x) is not odd, therefore f(x) is neither even nor odd}\)
Now, let \(g(x) = -2x-3 \), let check
\(g(-x) = 2x-3\neq g(x),~~hence,~~g(x) \text{is not even}\)
\(-g(x) = 2x+3 \neq g(-x),~~hence~~\text{g(x) is not odd, hence g(x) is neither even nor odd}\)
Now, combine \((f+g)(x) = f(x) +g(x) = x+3-2x-3 = -x\)
let check \((f+g)(-x) = x \\(-(f+g)(x) = -(-x) =x\)

hence \((f+g)(-x) = x=-(f+g)(x)\), therefore \((f+g)(x) \)is an odd function.