Inverse of a Multivariate FunctionDate: 05/30/2002 at 10:17:15 From: Subha Subject: calculus Let f: NxN->N such that f(x,y) = 2^x(2y + 1) - 1 for all natural numbers x, y. Let the inverse of f, g be given by g:N->NxN. Find g. Date: 05/30/2002 at 13:18:17 From: Doctor Peterson Subject: Re: calculus Hi, Subha. It's important to realize that we are talking about f: NxN->N. That is, first, we are dealing with natural numbers (in this case apparently taken to mean non-negative integers, judging by the behavior of the function), not real numbers; you will not want to use logarithms. Second, the inverse has to take a single natural number back to the PAIR of natural numbers (x,y) from which f created it; we can't just invert with respect to one variable as usual. So the question to ask is, how does f manage to take only one pair to any given natural number? The function is carefully designed. Note that x appears only in "2^x", which is even, while y appears only in "(2y+1)", which is odd, and has no factors of 2. As a result, there is no way to make the same product using a different x,y pair! Subtracting 1 from the product of these serves only to make the range all of N, since it takes (0,0) to 0 rather than to 1. So if you add the 1 back on, you can find x and y by factoring. Can you see how to do this? It will probably be hard to express the inverse as a formula, but you probably do not need to do that; an algorithm for finding g(z) should be sufficient. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 05/30/2002 at 18:01:53 From: Subha Subject: calculus How do I find the factor of f(x,y) by sustituting values for x and y? Please do tell me how to find the alogorithm to find g(x) which is the inverse of f(x,y). Date: 05/30/2002 at 20:45:06 From: Doctor Peterson Subject: Re: calculus Hi, Subha. Your function is f(x,y) = 2^x (2y+1) - 1 Given a value z, you need to find the pair (x,y) for which z = f(x,y) Let's just look at an example, and I'll let you decide how to describe the algorithm. Suppose z is 59. We want 2^x (2y+1) - 1 = 59 so we add 1 and have 2^x (2y+1) = 60 Now if we factor 60, we find that 2^x (2y+1) = 2*2*3*5 What power of 2 do you see here? What is x? What is the other (odd) factor? What is y in order for this to be 2y+1? Now write down a description of what you did to find x and y, and that will be the inverse function. One way you might express the inverse function is something like g(z) = (..., ....) where the "..." would be expressions involving z that give x and y. You might invent a function like even(x) = highest power of 2 that divides x and use that to write the expressions. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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