Inverse of a Function vs. Inverse Proportionality

Date: 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/