Where and why is identity function useful?

Question

I understand why function composition is important. It allows building large and complex functions from small and simple ones.

val f: A => B = ...
val g: B => C = ...

val h = f andThen g; // compose f and g

This composition conforms to identity and associativity laws.

Associativity is useful because it allows grouping f1 andThen f2 andThen f3 andThen f4 ... in any order. Now I wonder why identity is useful.

def f[T](t:T) = t   // identity function
val g: A => B = ... // just any function
g andThen f[B] == f[A] andThen g 

So, my question is where and why this identity useful.

Answer 1

The identity function is, well, the identity element for function composition, like 1 for multiplication or 0 for addition. We need it like we need 1 or 0. For instance, suppose we are writing a (higher-order) function that takes a list of functions and returns their composition: it is then natural to return the identity function for an empty list. This is just like we return 0 for an empty list if we are writing a function that takes a list of integers and returns their sum.