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

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mwmovr40 | College Teacher | (Level 1) Associate Educator

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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.

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