single transistor mixer
Yes, the output spectrum will be shifted by f at frequency domain.
For practical mixer implementation, one way is to use the non-linearity of certain devices as diodes or transistors to generate |n*f1±m*f2| where n, m are integers and f1, f2 are the required drequenciese ti be mixed together.
Another way to perform the mixing function is to use the carrier frequency to drive switches to reverse the polarity of the input signal.
For more information concerning mixer circuits you may refer to "RF microelectronics" by Behzad Razavi, chapter 6.