Answer
(a) and (d) all even,
(b) and (c) are odd.
Work Step by Step
We know that odd functions obey the principle:
$f(-x)=-f(x)$
and even functions follow the principle:
$f(-x)=f(x)$
(a) and (d) are even functions. This is because multiplying or dividing two odd functions makes them even (the negative sign cancels out).
$f(-x)g(-x)=-f(x)*-g(x)=f(x)g(x)$
Similarly, division will also lead to a negative sign cancellation.
(c) is an odd function. Consider:
$f(-x)-g(-x)=-f(x)-(-g(x))=-f(x)+g(x)=-(f(x)-g(x))$
Hence, it is odd based on the principles stated above.
(b) is an odd function. We can rewrite the function as follows:
$f(x)^3= f(x) \times f(x)^2$
Then
$f(-x)^3=f(-x)\times f(-x)^2=-f(x)\times f(x)^2=-f(x)^3$
Hence, it is odd based on the principles stated above.
