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### 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
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

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/
```
Associated Topics:
High School Definitions
High School Functions
High School Trigonometry

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