1. CS 250, Discrete Structures, Fall 2013
Lecture 2.4: Functions
CS 250, Discrete Structures, Fall 2013
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag

2. Course Admin HW1 Mid Term 1: Oct 8 (Tues)
Provided the solution soon
Done with grading
To be distributed today
Mid Term 1: Oct 8 (Tues)
Review Oct 3 (Thu)
Covers Chapter 1 and Chapter 2
HW2 coming out: early next week
Due Oct 15 (Tues)
2/24/2019
Lecture Functions

3. Outline Functions compositions common examples 2/24/2019
Lecture Functions

4. Function Composition
When a function f outputs elements of the same kind that another function g takes as input, f and g may be composed by letting g immediately take as an input each output of f
Definition: Suppose that g : A  B and f : B  C are functions. Then the composite
f g : A  C is defined by setting
f g (a) = f (g (a))
f g is also called fog
2/24/2019
Lecture Functions

5. Composition: Examples
Q: Compute gf where
1. f : Z  R, f (x ) = x 2
and g : R  R, g (x ) = x 3
2. f : Z  Z, f (x ) = x + 1
and g = f -1 so g (x ) = x – 1
3. f : {people}  {people},
f (x ) = the father of x, and g = f
2/24/2019
Lecture Functions

6. Composition: Examples
1. f : Z  R, f (x ) = x 2
and g : R  R, g (x ) = x 3
gf : Z  R , gf (x ) = x 6
2. f : Z  Z, f (x ) = x + 1
and g = f -1
gf (x ) = x (true for any function composed with its inverse)
3. f : {people}  {people},
f (x ) = g(x ) = the father of x
gf (x ) = grandfather of x from father’s side
2/24/2019
Lecture Functions

7. f n (x ) = f f f f  … f (x )
Repeated Composition
When the domain and codomain are equal, a function may be self composed. The composition may be repeated as much as desired resulting in functional exponentiation. The whole process is denoted by
f n (x ) = f f f f  … f (x )
where f appears n –times on the right side.
Q1: Given f : Z  Z, f (x ) = x 2 find f 4
Q2: Given g : Z  Z, g (x ) = x + 1 find g n
Q3: Given h(x ) = the father of x, find hn
2/24/2019
Lecture Functions

8. Repeated Composition A1: f : Z  Z, f (x ) = x 2.
f 4(x ) = x (2*2*2*2) = x 16
A2: g : Z  Z, g (x ) = x + 1
gn (x ) = x + n
A3: h (x ) = the father of x,
hn (x ) = x ’s n’th patrilineal ancestor
2/24/2019
Lecture Functions

9. Composition - a little theorem
Let f:AB, and g:BC be functions.
Prove that if f and g are one to one, then g o f :AC is one to one.
Recall defn of one to one: f:A->B is 1to1 if f(a)=b and f(c)=b  a=c.
Suppose g(f(x)) = y and g(f(w)) = y. Show that x=w.
f(x) = f(w) since g is 1 to 1.
Then x = w since f is 1 to 1.
2/24/2019
Lecture Functions

10. Commonly Encountered Functions
Polynomials:
f(x) = a0xn + a1xn-1 + … + an-1x1 + anx0
Ex: f(x) = x3 - 2x2 + 15; f(x) = 2x + 3
Exponentials:
f(x) = cdx
Ex: f(x) = 310x, f(x) = ex
Logarithms:
log2 x = y, where 2y = x.
2/24/2019
Lecture Functions

11. Some New functions Ceiling:
f(x) = x the least integer y so that x  y.
Ex: 1.2 = 2; -1.2 = -1; 1 = 1
Floor:
f(x) = x the greatest integer y so that x  y.
Ex: 1.8 = 1; -1.8 = -2; -5 = -5
Quiz: what is  1.1 ?
2/24/2019
Lecture Functions

12. Today’s Reading
Rosen 2.3
2/24/2019
Lecture Functions