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Lecture 2.4: Functions

Lecture 2.4: Functions. CS 250, Discrete Structures, Fall 2011 Nitesh Saxena * Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag. Course Admin. HW1 Due at 11am 09/09/11 Please follow all instructions Recall: late submissions will not be accepted

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Lecture 2.4: Functions

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  1. Lecture 2.4: Functions CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag Lecture 2.4 -- Functions

  2. Course Admin • HW1 • Due at 11am 09/09/11 • Please follow all instructions • Recall: late submissions will not be accepted • Mid-Term 1 on Thursday, Sep 22 • In-class (from 11am-12:15pm) • Will cover everything until the lecture on Sep 15 • No lecture on Sep 20 • As announced previously, I will be traveling to Beijing to attend and present a paper at the Ubicomp 2012 conference • This will not affect our overall topic coverage • This will also give you more time to prepare for the exam Lecture 2.4 -- Functions

  3. Course Admin • HW1 grading potentially delayed • TA/grader is sick with chicken pox • We will try to finish it up as soon as possible. Apologies for the delay. • In any case, HW1 solution will be released in a few days from now. So, you can prepare for your exam without any interruptions Lecture 2.4 -- Functions

  4. Outline • Functions • compositions • common examples Lecture 2.4 -- Functions

  5. 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 Lecture 2.4 -- Functions

  6. 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 Lecture 2.4 -- Functions

  7. 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 Lecture 2.4 -- Functions

  8. 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 f4 Q2: Given g: Z  Z, g (x ) =x + 1 find gn Q3: Given h(x ) = the father of x, find hn Lecture 2.4 -- Functions

  9. Repeated Composition A1: f: Z  Z, f (x ) =x 2. f4(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 Lecture 2.4 -- Functions

  10. Composition - a little problem 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. Lecture 2.4 -- Functions

  11. 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. Lecture 2.4 -- Functions

  12. 0 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 -1.2 + 1.1 ? Lecture 2.4 -- Functions

  13. Today’s Reading • Rosen 2.3 Lecture 2.4 -- Functions

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