"People who don't count won't count." - (Anatole France, 1844-1924) "Mighty is geometry; joined with art, resistless." - (Euripides, 480 - 406 B.C.) Arthur Cayley, 1821-1895: An English mathematician who founded the theory of quantics (algebraic functions with two or more variables), Cayley also devised a noncommutative matrix algebra and contributed significantly to quantum mechanics and the theory of manifolds. Cayley was one who believed mathematics is embedded in man's mind, as a contrast to others (Plato, for instance) who saw it as existing in a world outside of man. In an address to the British Association for the Advancement of Science (1883), Caley said that "We are...in possession of cognitions, a priori, independent, not of this or that experience, but absolutely so of all experience. These cognitions are a contribution of the mind to the interpretation of experience."

When Herkimer lived in a grass hut, why wouldn't he allow a friend to store a royal chair in his dwelling? Answer: He was told that people who live in grass houses shouldn't stow thrones. Herky wants to know: When cheese gets its picture taken, what is it supposed to say? Since croutons are stale bread to begin with, why do they come in airtight packages?

ASSIGNMENT #68 Reading: Review Section 9.2, as necessary. Written: On the graph paper provided, provide a very neat graph for problems 27, 29, 33, 35, and 37 on page 544. Clearly identify all asymptotes and intercepts, and state domain and range for the function.

Mathematical word analysis: PROPORTION: From the Greek proportione (referring to one's share or quota). If, for instance, you divide an apple pie equally among 8 people, the proportion 1/8 represents each individal's share of the pie.

Consider a function like f(x) = y = x/(4x+3). Here are things that can be observed about the function.

The function is not defined if 4x+3 = 0. That is, the number -3/4 is not in the domain of the function. The domain is all real numbers except -3/4. The line x = -3/4 is an asymptote.

As x--> infinity, y -->1/4.

As x --> -infinity, y --> 1/4.

The value of y can never equal 1/4. The range is all real numbers except 1/4.

The value of y gets indefinitely large or indefinitely small as x gets very close to -3/4.

f(0) = 0. The graph goes through the origin.

With this information, you should be able to draw a reasonable sketch of the graph of f. Your calculator can be helpful, but it is limited in terms of what it can actually show.

Problem: What is the domain and range of the function y = (x+7)/(x+8)? Solution (with communication): The denominator x + 8 cannot be zero. Hence, the domain is all real number except -8. Since the numerator is always one less than the denominator, the value of y cannot be 1. The range is all real numbers except 1.

Problem: Given the function y = (x+7)/(x+8). As indicated in the problem at the left, the value of y cannot be 1. Find the value of x for which y = 0.99. Solution (with communication): y = 0.99 ==> 0.99 = (x+7)/(x+8) ==> x+7 = 0.99(x+8) = 0.99x+7.92 ==> .0.01x = 0.92 ==> x = 0.92/0.01 = 92.