The frequency of sounds that the source emits doesn't actually change. Consider the analogy below.
Suppose a boy throws one ball in every second towards the teacher. Assume that the balls travel with same velocity. If the thrower is stationary, the teacher will receive 1 ball/Second. If the thrower is moving towards the teacher, she will receive balls more frequently as the balls will be less spaced out. The inverse is true if the thrower is moving away from the teacher. So it's actually the wavelength which is affected and as a result the received frequency will also get affected.
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