Inverse of a Function vs. Inverse ProportionalityDate: 11/20/2002 at 08:29:33 From: Mark Kalin Subject: Inverse of a function vs. inverse proportionality Hi, I'm currently teaching a grade 12 course that covers the topics of inverse of a function and inverse proportionality. Finding the inverse of a function and graphing yields a graph that has been reflected in the line y = x relative to the function. Inverse proportionality, however, yields a reciprocal relation graphically. Why do these two things have similar names yet mean different things? Also, inverse notation employs the use of a superscipt negative one, which of course as a math teacher I know does not mean the reciprocal but the inverse. Why such confusing terminology for students? Or am I missing something here? Date: 11/20/2002 at 08:37:50 From: Doctor Jerry Subject: Re: Inverse of a function vs. inverse proportionality Hi Mark, When you compose a function and its inverse, f o f^{-1} you get (just manipulating symbols) f^0, which might be regarded as the identity function. This is just a reason for the -1 notation. Rarely do any of us get confused about this. However, I use arcsin instead of sin^{-1} for the inverse sine function. Inversely proportional is an ancient term and in some sense turns things around; as one thing gets bigger, the other gets smaller. The inverse of a function also turns things around, but in a different sense. So, the two usages are related in a general kind of way, but not in a mathematical way. There are many instances in mathematics and elsewhere of conflicting usages. Most of us come to accept this as something not worth worrying about. Best wishes, - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 11/20/2002 at 09:07:24 From: Doctor Peterson Subject: Re: Inverse of a function vs. inverse proportionality Hi, Mark. Both usages fall under the general meaning of "inverse," which my dictionary gives as "reversed in order, nature, of effect; turned upside down; inverted." In the one case, we are reversing the action of a function, putting something in the output, so to speak. In the other, we are turning a ratio upside down. So it makes sense to use the same word in both places. This is just the nature of English, or any other language. Look in your dictionary, and you will find innumerable cases of words that are used in very different ways, even though they start with the same root meaning, because they found their way into different contexts. Occasionally this can cause confusion, as in accidentally ambiguous statements or deliberate puns; but usually the meaning is clear from the context. And we use words like this with a natural, visual meaning, rather than make up entirely new words, because they communicate clearly. There are different ways in which things are turned upside down, and in each case if you know both this basic meaning of the word and the particular context, the meaning of the word in context is clearer than if we invented a whole new word. As for the symbol for the inverse of a function, that arises out of analogy. We can consider functions as objects in themselves, and combine them by the operation of composition. If we write that as if it were multiplication, fg, then it is natural for the inverse function to be written as if it were a reciprocal, since f f^-1 = i and f^-1 f = i. This sort of generalization is the foundation of abstract algebra, and the inverse notation is standard there. The only place this really causes confusion is in trigonometry, where an old convention allows us to put exponents on the function rather than the whole expression (sin^2 x rather than (sin x)^2). In this case, it is not clear whether sin^-1 should mean 1/sin or the inverse function. For that reason (and because in computer programming we use "asin"), I use arcsin rather than sin^-1. It would be better to drop the convention of sin^2, but that is too well established. Notation has developed in a haphazard way, just like English, and not everything makes sense! Here are some relevant pages from our archives: Trigonometry Terminology http://mathforum.org/library/drmath/view/54185.html Inverses http://mathforum.org/library/drmath/view/54597.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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