Design learning system steps 2 and target function in machine learning

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If I understand your question correctly then the target function is a function that people in Machine learning career tend to name it as a hypothesis.

In machine learning, you are given a lot of data and its target value (sometime called class or label or answer).

Specifically, the target function can only be inferred in a Supervised learning task.

A target function maps data to its target value such as if you have picture of a digit, the function then receives that picture as the input and spit the digit value as the output.

The process of finding that function is called learning or training.
Target function, you can imagine it as the true predictor or a God-given Model. It can predict with 100% accuracy on the future data.

This is a function that knows and maps a full-relationship of the features/input variables to the Response/Output variable. Or, you can just treat it as imaginary function that always correct at every predictions.

In a (supervised) machine learning, your task is to select a function from the uncountable hypothesis functions that is closest to the target function.