Assignment 52

 "Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, are are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity." -- (Mary Somerville, 1780-1972) David Hume (Scotland, 1711-1776): One of a number of eighteenth-century philosophers who did not believe in the doctrine of an external world following fixed mathematical laws. As did Immanuel Kant (1724-1804), Hume claimed that mathematics is not inherent in the physical world but comes from the human mind. The intuitionist school of mathematical thought that was founded by Dutch mathematician Luitzen Brouwer (1881-1966) built their beliefs around the philosophies of Hume and Kant. In a nutshell, the intuitionists believe that mathematics is a human activity that originates and thrives in the mind, and that it is independent of the real world. While knowledge and understanding may begin with experiences, they do not originate from experience. So, for example, a logical construction of the real number system is not acceptable. What did Herkimer do when the police where chasing him for stealing a set of bathroom scales? Answer: He stopped and stepped on the scales so he could get a weigh. Herky's friends: ALLISON WONDERLAND...she loved the stories written by Lewis Carroll. EDDIE TORIAL...this guy wrote a daily newspaper column. ASSIGNMENT #52 Reading: Section 7.5, pages 431-434. Written: Pages 434-435/23-41 (odds). In each case, sketch a rough graph of the function and clearly state the domain and range of the function.

 Mathematical word analysis:QUOTIENT: From quotcient , a Middle English word meaning "number of times." A quotient indicates a division, and when you divide you determine the number of times the denominator is contained in the numerator.
Mathematics is a language. Terminology is important. As with any language, if you don't read it properly, or if you don't know what the words mean, it can be difficult.

A characterization of a function is that it is a set of ordered pairs. The domain of a function is the set of all x values, and the range is the set of all y values. If you think of a function as a rule, then the domain is the set of numbers on which the rule operates, and the range is the set of numbers produced by the rule.

As a simple example, consider y = f(x) = x2. Basically, f is a rule that says "take a number and square it." Now, this rule can operate on any real number, so the domain of f is the set of real numbers. Note, however, that the rule cannot produce negative numbers. If can produce 0 and all positive numbers. So, the range of f is all non-negative numbers. We could also describe the range by writing 0 £ y.

Now consider y = f(x) = (x - 7)1/2 + 5. If we know the language of mathematics, we realize that (x-7)1/2 represents the square root of (x-7). In the real number system, we cannot take the square root of negative numbers. Hence, the domain of f is 7 £ x. Since (x-7)1/2 cannot be negative, the function f can only produce values that are 5 or greater. Hence, the range of f is 5 £ y. If we think of f as a rule, then the rule f says "take a number, subtract 7, take the square root, then add 5. The rule cannot operate on values less than 7, and it cannot produce values less than 5.

MATH IS A LANGUAGE: It is a very logical language that makes total sense if you will learn to read it properly.

 Problem: State the domain and range of the function y = (x - 12)1/2 + 23. Solution (with communication): (x - 12)1/2 is not defined in the real number system if x is less than 12. Also, (x - 12)1/2 cannot be negative, but can be zero when x = 12. Hence, the domain of the function is all real numbers equal to or greater than 12, and the range of the function is all numbers equal to or greater than 0 + 23 = 23. Problem: Find the domain and range of the function y = (x - 5)1/2 + (5 - x)1/2 Solution (with communication): We must have 0 £ x - 5 and 0 £ 5 - x. Hence, we must have 5 £ x and x £ 5. The only solution to this system is x = 5. The graph of this function is the single point (5,0). Hence, the domain is {5} and the range is {0}.