Odd or even functions refer to the nature of the symmetry of the function. If one switches the sign of the value substituted into the function:replacing "x" with "-x" (eg: replacing 1 with -1) one of three things can happen to the result of f(x):
the value will remain the same as for the original value of x
the value will be the opposite of the value generated by the original value of x
the value will be neither the original value nor its opposite.
If the value generated by the opposite of x is the same as for x, the function is said to be even.
If the value generated by the opposite of x is the opposite of that for x, the function is said to be odd.
In the case of f(x) =xsin(x) can see that sin(x) and sin(-x) produce the same values. However, x and -x produce opposite values. Therefore
f(x) = -f(-x) which is the requirement of being odd.
f(x) = xsin(x) is odd.