"I know numbers are beautiful. If they aren't beautiful, nothing is." - (Paul Erdos) Henri Poincare (France, 1854-1912): Poincare believed that mathematics can be made to fit physical reality. His 491 publications enrich all branches of mathematics. He was the first to use topological methods in algebraic geometry. He is generally considered to be the founder of modern topology. Topology is an interesting and extremely useful branch of modern mathematics that deals with the aspects of a surface's nature which is unaffected by the deformation of the surface. Poincare once said "If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science."

When Herkimer was a policeman, why didn't he arrest the judge who was wearing a stolen raincoat? Answer: He believed that you should not book a judge by his cover. Herky wants to know: If 21 is pronounced twenty-one and 31 is pronounced thirty-one, why isn't 11 pronounced onety-one? Why are a wise man and a wise guy opposite things?

ASSIGNMENT #67 Reading: Section 9.2, pages 540-542 Written: Page 544/25,27,29,33,35,37,44,45. In each case, sketch a neat graph. Identify domain and range for each function. Also, identify any asymptotes that exist. (Think before you look at the graph on your calculator. What do you expect to see?)

Mathematical word analysis: POSITIVE: From the root posit, which means to place or to set. Items were placed (or set) and then counted using what we know as positive integers.

Consider a function y = f(x) = 4/(x-3) + 12.

Since division by 0 is impossible, it follows that x = 3 cannot be in the domain of f. However, the function is clearly defined for all other real numbers, so the domain of f consists of all real numbers except 3.

Let's do a bit of thinking. If x is close to 3, say 3.1, then f(3.1) = 4/(.1) + 12 = 40 + 12 = 52.

Suppose x = 3.01. Then f(3.01) = 4/(.01) = 400 + 12 = 412.

Suppose x = 3.000001. Then f(3.000001) = 4/(.000001) + 12 = 4,000,000 + 12 = 4,000,012.

Suppose x = 2.99999999. Then f(2.99999999) = 4/(-.00000001) + 12 = -400,000,000 + 12 = -399,999,988.

If is obvious that as x gets very close to 3, then y gets very large or very small, depending upon whether 3 is being approached from the right or from the left. And note that since 4/(x-3) can never be 0, it follows that y can never assume the value of 12.

For this function, the domain is all real numbers except 3, the range is all real numbers except 12, the line x = 3 is a vertical asymptote, and the line y = 12 is a horizontal asymptote.

To get a good graph of your calculator, it might be useful to use the TABLE feature. Put the function in the Y= editor, then go to Table Set (TBLSET), and look at values of y for various values of x. For the particular function y = 4/(x-3)+12, you will find that the y values are fairly close to 12 everywhere except in the region around x = 3. In other words, one would certainly be wise to make sure y = 12 is included in your WINDOW.

Problem: Identify the domain and range of the function y = 12/(15-x) + 22. Also identify asymptotes. Solution (with communication): The only restriction on x is that 15-x cannot be 0. Hence, the domain is all real numbers except x = 15. The only value that 12/(5-x) cannot assume is 0. Hence, the range of this function is all real numbers except y = 15. x = 15 is a vertical asymptote. Y = 22 is a horizontal asymptote.

Problem: Create a function that has x = 99 as a vertical asymptote and y = 10 as a horizontal asymptote. Solution (with communication): There are numerous possibilities, but a function satisfying the stated conditions is y = 1/(x-99) + 10.