Is f(x)=xsinx an odd or even function?
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.