From jacobson@math-cs.cns.uni.edu Thu Nov 8 21:10:58 2007 Date: Thursday, 8 November 2007 9:10 P.M. From: Mark Jacobson To: 810-080-02-fall@uni.edu Subject: Page 277, Exercise 2B... Page 277 of textbook, problem 2B. 2. All functions in this problem have the set of real numbers R as their domain and codomain. 2 (b) If f(x) = 3x - 2 and (g o f)(x) = 9x - 9x then what is function g(x)? (g o f)(x) = g( 3x - 2 ) since f(x) = 3x - 2 We know that 2 g( 3x - 2 ) = 9x - 9x Clearly, g(x) will have to involve squaring, i.e. 2 g(x) = x + ???? Why? 2 To get 9x from an x input involves squaring, and indeed, with 3x as one of the input terms to function g(), its 2 clear that g(x) = x + someYetToBeDeterminedLowerOrderTerms 2 2 is on the right track. [ Note: (3x) = 9x ] 2 Now the question is, what does that g(x) = x + mysteryTerms highest order term do all by itself to the output of f(x), i.e. what does it do to (3x - 2)? It squares it, of course. ------------- 2 2 (3x - 2) = (3x - 2)(3x - 2) = 9x - 12x + 4 ------------- Now, notice that -12x needs to have 3x added to it, so that we would have (-12x + 3x) = -9x, right? Now, where could we find a 3x? (3x - 2) without squaring is just 3x - 2, i.e. its a lower order term in the polynomial, since 1 2 (3x - 2) is (3x - 2) and is lower degree than (3x - 2) 1 2 x is lower degree than x squared, i.e. x is lower degree than x Aha, we now know that g(x) has to square its input and add its input to the result, i.e. 2 we now know that g(x) = x + x + someMysteryTerm(possibly 0) 2 2 (3x - 2) + (3x - 2) + ????? = 9x - 9x 2 2 9x - 12x + 4 + 3x - 2 + ????? = 9x - 9x 2 2 9x - 12x + 3x + 4 - 2 + ????? = 9x - 9x 2 2 9x - 9x + 2 + ????? = 9x - 9x So no we have the final piece to the puzzle. ????? has to be -2, right? So, g(x) does what to its input? Squares it, adds it and subtracts 2 from it. 2 g(x) = x + x - 2 Just to double check now, with f(x) = 3x - 2, then g o f(x) = g(3x - 2) ---------------- -------- --- and g(3x - 2) = (3x - 2)(3x - 2) + (3x - 2) - 2 ---------------- -------- --- A B C 2 A = 9x - 12x + 4 B = 3x - 2 C = - 2 ----------------- 2 2 9x - 9x + 0 = 9x - 9x Whew, that indeed was a tough problem. There will be nothing that tough about composition of functions on the exam, but understanding the above problem now probably makes the simpler problems of f o g(x) very understandable. In computer science, there is the concept of reverse engineering executable software to come up with the Pascal or C++ or Ada or C or Lisp or Scheme or COBOL or Fortran or PERL code that it was compiled from. This, in some ways, is analogous to that process with functions. Here is function f Here is function g What is function f o g? Or what is f o g(3)? Very simple problems compared to: Here is function f Here is function g o f Figure out what function g had to be!!!! Sorry I did not preview this problem ahead of time or discover after it was assigned that it should be cancelled or else an explanation of a similar problem like the explanation above should have been sent out last week! Transitivity analogy, plus, if transitivity could be quantified: UNI football beats Missouri State by 21 points (38-17). Missouri State beats Indiana State by 56 points (63-7). Does this mean UNI football wins over Indiana State by 77 points? Or just that (UNI beats Missouri State and Missouri State beats Indiana State) implies UNI beats Indiana State Or none of the above, since on any given day ... ? Mark