What is the apply function in Scala?

It comes from the idea that you often want to apply something to an object. The more accurate example is the one of factories. When you have a factory, you want to apply parameter to it to create an object.

Scala guys thought that, as it occurs in many situation, it could be nice to have a shortcut to call apply. Thus when you give parameters directly to an object, it's desugared as if you pass these parameters to the apply function of that object:

class MyAdder(x: Int) {
  def apply(y: Int) = x + y
}

val adder = new MyAdder(2)
val result = adder(4) // equivalent to x.apply(4)

It's often use in companion object, to provide a nice factory method for a class or a trait, here is an example:

trait A {
  val x: Int
  def myComplexStrategy: Int
}

object A {
  def apply(x: Int): A = new MyA(x)

  private class MyA(val x: Int) extends A {
    val myComplexStrategy = 42
  }
}

From the scala standard library, you might look at how scala.collection.Seq is implemented: Seq is a trait, thus new Seq(1, 2) won't compile but thanks to companion object and apply, you can call Seq(1, 2) and the implementation is chosen by the companion object.

Mathematicians have their own little funny ways, so instead of saying "then we call function f passing it x as a parameter" as we programmers would say, they talk about "applying function f to its argument x".

In mathematics and computer science, Apply is a function that applies functions to arguments.
Wikipedia

apply serves the purpose of closing the gap between Object-Oriented and Functional paradigms in Scala. Every function in Scala can be represented as an object. Every function also has an OO type: for instance, a function that takes an Int parameter and returns an Int will have OO type of Function1[Int,Int].

 // define a function in scala
 (x:Int) => x + 1

 // assign an object representing the function to a variable
 val f = (x:Int) => x + 1

Since everything is an object in Scala f can now be treated as a reference to Function1[Int,Int] object. For example, we can call toString method inherited from Any, that would have been impossible for a pure function, because functions don't have methods:

  f.toString

Or we could define another Function1[Int,Int] object by calling compose method on f and chaining two different functions together:

 val f2 = f.compose((x:Int) => x - 1)

Now if we want to actually execute the function, or as mathematician say "apply a function to its arguments" we would call the apply method on the Function1[Int,Int] object:

 f2.apply(2)

Writing f.apply(args) every time you want to execute a function represented as an object is the Object-Oriented way, but would add a lot of clutter to the code without adding much additional information and it would be nice to be able to use more standard notation, such as f(args). That's where Scala compiler steps in and whenever we have a reference f to a function object and write f (args) to apply arguments to the represented function the compiler silently expands f (args) to the object method call f.apply (args).

Every function in Scala can be treated as an object and it works the other way too - every object can be treated as a function, provided it has the apply method. Such objects can be used in the function notation:

// we will be able to use this object as a function, as well as an object
object Foo {
  var y = 5
  def apply (x: Int) = x + y
}


Foo (1) // using Foo object in function notation 

There are many usage cases when we would want to treat an object as a function. The most common scenario is a factory pattern. Instead of adding clutter to the code using a factory method we can apply object to a set of arguments to create a new instance of an associated class:

List(1,2,3) // same as List.apply(1,2,3) but less clutter, functional notation

// the way the factory method invocation would have looked
// in other languages with OO notation - needless clutter
List.instanceOf(1,2,3) 

So apply method is just a handy way of closing the gap between functions and objects in Scala.