====== Lab 2. High order functions ======

Objectives:
  * implement and use **higher-order** functions. A **higher-order** function takes other functions as parameter or returns them
  * implement **curry** and **uncurry** functions, and how they should be properly used (review lecture).

** Create a new Scala worksheet to write your solutions **

===== 2.1 Intro. Functions as parameters =====
**2.1.1** Write a function ''apply'' that takes an integer and return the result of the applied function on the given integer. Start from the code stub below:
<code scala>
def apply(n: Int, f: Int => Int): Int = {
   ???
}
</code>

**2.1.2** Write a function ''doubler'' that returns a function that doubles the input it receives (an integer). Start from the code stub below:
<code scala>
def doubler(): Int => Int = {
   ???
}
</code>

===== 2.2 Custom high order functions =====
**2.2.1** Define the function ''foldWith'' which uses an operation ''op'' to reduce a range of integers to a value. For instance, given that ''op'' is addition (+), the result of folding the range 1 to 3 will be 1+2+3=6. ''foldWith'' should be curried (it will take the operation and return another function which expects the bounds).
<code scala>
def foldWith (op: (Int,Int) => Int)(start: Int, stop: Int): Int = {
  def tail_fold(crt: Int, acc: Int): Int  = ???
  ??
}
</code>

**2.2.2** Define the function ''foldConditional'' which extends ''foldWith'' by also adding a predicate ''p: Int => Int''. ''foldConditional'' will reduce only those elements of a range which satisfy the predicate.

<code scala>
def foldConditional(op: (Int,Int) => Int, p: Int => Boolean)(start: Int, stop: Int): Int = ???
</code>

**2.2.3** Write a function ''foldMap'' which takes values $math[a_1, a_2, \ldots, a_k] from a range and computes $math[f(a_1)\;op\;f(a_2)\;op\;\ldots f(a_k)].
Use the ''apply'' and ''foldWith'' methods
<code scala>
def foldMap(op: (Int,Int) => Int, f: Int => Int)(start: Int, stop: Int): Int = ???
</code>

===== 2.3 Curry vs Uncurry =====
**2.3.1** Modify the function below so that it's curry and use it to calculate ''5*3''
<code scala>
def multiply(x:Int, y:Int): Int => x * y
</code>

**2.3.2** Modify the function below so that it's curry and use it to compare 3 numbers and return the maximum
<code scala>
def compare(x: Int, y: Int, z: Int): Int =
{
  if x > y && x > z then 
    x
  else if y > x && y > z then
    y
  else
    z
}
</code>

===== 2.4 Function transformations =====
The graph of a function can undergo different geometric transformation such as scaling, shifting, rotating, mirroring and so on. The result of those transformation will also be a function that looks similarly to the original. In this exercice we will particularly work with lines. A line is a linear equation of the form $math[f(x) = a*x + b]

**2.4.1** Implement a function that shifts a line on Oy axis by a certain amount $math[\Delta y]
<code scala>
def shiftOY(line: Double => Double, delta_y: Double): Double => Double = {
   ???
}
</code>

**2.4.2** Implement a function that shifts a line on Ox axis by a certain amount $math[\Delta x]
<code scala>
def shiftOX(line: Double => Double, delta_x: Double): Double => Double = {
   ???
}
</code>

**2.4.3** Implement a function that checks if two lines intersect at an integer value from a given interval
<code scala>
def intersect(line1: Double => Double, line2: Double => Double)(start: Int, stop: Int): Boolean = {
   ???
}
</code>