Description

Please do not write on this sheet of paper
And do not use laptops during the session
Function values

This week we will work on playing with functions as values.
Ex 1.

Define the function flip. It takes a function and returns the same function, but with the arguments flipped.

def flip(f: (Int, Double) => Int): (Double, Int) => Int = ???

Ex 2.1

Define the identity function for integers, which, given an Int , returns it

val id: Int => Int = ???

Ex 2.2

Define the compose function, that, given 2 functions f, g, returns a function that composes them, i.e., f ∘ g.
def compose(f: Int => Int, g: Int => Int): Int => Int = ???

What does compose(id, f)(k) evaluate to, for some function f and integer k ?

Ex 2.3

Define the function repeated, which takes a function and repeatedly applies it n times (n ≥ 0).

def repeated(f: Int => Int, n: Int): Int => Int = ???

Hint: What values should be returned by repeated(x => x + 1, 0) and repeated(x => x + 1, 3)
?

Ex 3.
Write a function fixedPoint with the following signature:

def fixedPoint(f: Int => Int): Int => Int

The function takes a function f and returns a function that applies f up until it reaches a fixed point. A value x is a fixed point of f if f(x) == x.
For each of the following expressions, indicate whether it terminates, and if so, what is the value returned:
– fixedPoint(id)(123456)
– fixedPoint(x => x + 1)(0)
– fixedPoint(x => if (x % 10 == 0) x else x + 1)(35)
– fixedPoint((x: Int) => x / 2 + 5)(20)

Ex 4.1

Define the function curry2, that curries a two arguments function.

def curry2(f: (Int, Int) => Int): Int => (Int => Int) = ???

Hint: what should curry2((x, y) => x + y)(1) return?

Ex 4.2

Define the function uncurry2. It takes a curried function, and creates a two-argument function.

def uncurry2(f: Int => Int => Int): (Int, Int) => Int = ???

Ex 5.1

Write the sum function with the following signature:

def sum(a: Int, b: Int)(f: Int => Int): Int = ???

b−1
Which returns the following value: i∑=a f(i)
Bonus point: Can your implementation be tail recursive ?

Ex 5.2

Write the quadratic function with the following signature:
def quadratic(c: Int): Int => Int = ???

Which returns a function that takes an integer x as argument and returns (x − c)2 . Ex 5.3

Using the above functions, define the function quad3Integrate which, given two integers a and b, b−1
∑ (i − 3)2 outputs the following value: i=a

def quad3Integrate(a: Int, b: Int): Int = ??? val quad3Integrate: (Int, Int) => Int = ???