what is the difference between being neither even nor odd and being none of these

Question

$f(x)$ is an odd function and $g(x)$ is neither even nor odd,Then $f(x) + g(x)$

(a). Even

(b) Odd

(c)Nither even nor odd

(d)None of these.

$\boldsymbol{My}$$\boldsymbol{Approach}$$\Longrightarrow$I know (a) and(b) incorrect.

But i dont know which one is correct between (c) and (d).What is the difference between (c) and (d).??

And What is the key to recognize ???

Please explain in detail.

Answer 1

Let $f(x) = 1$. Then $f\;$is even.

First, let $g(x) = x-1$. Then $g\;$is neither even nor odd, but $f+g\;$is odd.

Next, let $g(x) = x+1$. Then $g\;$is neither even nor odd, and $f+g\;$is also neither even nor odd.

The first example disqualifies $(c)$, and the second example disqualifies $(a)\;$and $(b)$.

Hence, "none of these" is correct.