## Calculus (3rd Edition)

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.