Transfer function sinusoidal response
For any LTI (Linear Time Invariant) system (considering the steady state response i.e. ignoring the transient response which generally becomes negligible in a very short period), if the input is a sinusoidal signal, the output is always a sinusoidal signal of the same frequency.  But the amplitude of the output sinusoidal can be different from the input amplitude.  And the ratio of the output amplitude to input amplitude is a constant for that particular frequency - whatever the input amplitude and phase are. Similarly, the difference between the input and output sinusoidal phases is also constant for the particular frequency - irrespective of the amplitude. The transfer function can be used to give this input-output amplitudes ratio and input-output phase difference as a function of frequency.  Hope this helps.  The frequency dependency is derivable from the input/output differential equation in analog case or difference equation in discrete case.