One of the core ideas of this chapter is rather simple: in order to predict the mean of an output variable, we can apply an arbitrary function to a linear combination of input variable.
Where 
is a function, we will call inverse link function. There are many inverse link functions we can choose; probably the simplest one is the identity function. This is a function that returns the same value used as its argument. All models from Chapter 3, Modeling with Linear Regression used the identity function, and for simplicity we just omit it. The identity function may not be very useful on its own, but it allows us to think of several different models in a more unified way.
Why do we call f, the inverse link function, instead of just the link function? Because traditionally people apply functions to the other side of equation 4.1, and unfortunately for us,...