Assignment 50

"The way his horses ran could be summed up in a word. Last. He once had a horse who finished ahead of the winner of the 1942 Kentucky Derby. Unfortunately, the horse started running in the 1941 Kentucky Derby." -- (Groucho Marx)

Math History Tidbit:

John Napier (1550-1617): A native of Scotland, Napier invented logarithms as a laborsaving device to speed up numerical computation processes. His invention was enthusiastically adopted throughout Europe. Today we consider a logarithm as an exponent, but Napier never thought of his discovery in this way. The modern use of logarithms in high-powered mathematics course and in computer technology would probably boggle Napier's mind. In 1614 Napier published a book titled A Description of the Wonderful Law of Logarithms. It was enthusiastically accepted throughout Europe as the use of the tables it contained enabled tedious multiplication and division operations to be done as easily as addition and subtraction. Interestingly, Napier, like Michael Stifel (1486-1567) before him, was a number mystic, and spent time attempting to determine a date upon which the world would end.

Herkimer's Corner

When Herkimer ran a casino, what did he tell the wife of a doctor who wanted to have her husband paged?

Answer: "The house does not make doctor calls."

Things Herky would like to know:

Are there turtles that wear people-neck sweaters?

If a minister delivered a sermon that he simply made up as he was delivering it, would it be correct to say the he doesn't practice what he preaches?


Reading: Section 7.4, pages 422-425.

Written: Page 426/14-24 (Write up neatly. Don't get sloppy.)

Items for reflection:

Mathematical word analysis:
SCALENE: From the Greek word scalenos, which means "uneven." A scalene triangle has uneven sides (sides of unequal length).

There are very important mathematical concepts in this section, including the inverse of a function. Basically, a function can be thought of as a rule that takes x and produces a unique value y. The inverse of a function must be a rule that tells you how to get y back to x, where it came from. The inverse of a function is not necessarily a function. If f is a function, then its inverse is designated by f-1. OK, here is new language. If f is a function, then f-1 does not mean 1/f. Don't make this complicated. Learn the language.

If y = f(x) = 3x-2, then f is a rule that says "take a number, multiply it by 3, then subtract 2." OK, then f-1 must be a rule that says "take a number, and 2, then divide by 3." That is, f-1(x) = (x+2)/3.

Let's check this out. We have f(10) = 3(10) - 2 = 28. OK, then the inverse rule must tell you what to do to 28 to get back to 10. We have f-1(28) = (28+2)/3 = 30/3 = 10.

Also note that if f(x) = 3x-2, then the inverse function should tell us what to do with 3x-2 to get back to x. We have f-1(3x-2) = [(3x-2)+2]/3 = 3x/3 = x.

A function is simple a collection of ordered pairs. To get the inverse of a function, you need only interchange x and y.

Example: If f = {(3,5), (44,99)}, then f-1 = {(5,3), (99,44)}.

Example: If y = 3x - 2, then we can get the inverse function by interchanging x and y, then solving for y. If we took this approach, we would get x = 3y - 2 ==> 3y = x + 2 ==> y = (x+2)/3, which jives with what the rule approach got us. (Check above.)

Your calculator can graph the inverse of a function. You need the DRAW menu. If you have a function Y1 defined, then DrawInv Y1 will produce the inverse function on the screen. Since the inverse of a function is not necessarily a function, you can't produce the inverse directly from the Y= menu. For instance, the inverse of y = x2 is not a function. Think about this. If a rule says "take a number and square it," this rule is a function. The inverse rule would say "take a number and produce its square root." Now, a positive number has two square roots, so the inverse rule is not a function.

Problem: If f = {(3,33), (4,44), (5,55),(6,66)},
what is f-1?

Solution (with communication): To get f-1, merely interchange x and y. Hence

f-1 = {(33,3),(44,4),(55,5),(66,6)}.


Problem: If f(t) = 2t - 10, what is f-1(a+b)?

Solution (with communication):

f is a rule that says "take a number, multiply it by 2, then subtract 10." Hence f-1 must be a rule that says "take a number, add 10, then divide by 2." Hence

f-1(x) = (x+10)/2. It follows that

f-1(a+b) = (a+b+10)/2.