The correct option is C I, II and III
Let f(x)=ax and g(x)=−bx
Both are injective, functions
Now for a special case where a=b we get f(x)+g(x)=0, hence a constant function.
Therefore, A is incorrect.
Now we know that functions of type y=x2 or y=−x2 are many-one functions.
That is f(x1)=f(x2)
→x1=±x2 ...(hence many on one function).
f(−x)=−f(x), since sin(x) is odd function.
However it is periodic with a period of 2π
Hence it is not one on one function.
Hence all of the above are incorrect.