Inverses and Reciprocals of Functions

Date: 09/28/2005 at 14:26:03
From: Anthony
Subject: When is F inverse (x) the same as 1/F(x)

I'm confused about when a negative one exponent means reciprocal and
when it means inverse, particularly with trig functions.

For example, x^(-1) means 1/x, but sin^(-1)(x) does not mean 1/sin(x).

Date: 09/28/2005 at 19:48:46
From: Doctor Vogler
Subject: Re: When is F inverse (x) the same as 1/F(x)

Hi Anthony,

Thanks for writing to Dr. Math.  That's a good question:  When you
raise a function to the -1 power, what does that mean?  The short
answer is that it means "the inverse," but, unfortunately, there are
at least two meanings for that.  See also

  Inverses
    http://mathforum.org/library/drmath/view/54597.html 

When you write

  x^(-1),

or

   -1
  x  ,

that is, x raised to the -1 power, that is the same as the 
multiplicative inverse of x, or 1/x.  So if you write

  f(x)^(-1),

or

      -1
  f(x)  ,

then generally you mean 1/f(x), the multiplicative inverse of the
number f(x).  Similarly, if you write

  f(x)^2,

or

      2
  f(x)

you generally mean the square of the number f(x).

By contrast, it is more common for the inverse function to be written

  f^(-1)(x),

or

   -1
  f  (x),

with the exponent on the f and before the parentheses.  The inverse
function is different from the multiplicative inverse and has nothing
to do with 1/f(x).

So why do they use a power of -1 to mean inverse function?  The main
reason is that when people deal with *iterating* a function, they
often write

  f^n

or

   n
  f

to mean f *composed* with itself n times (rather than f *multiplied*
by itself n times).  So then

   2
  f (x) = f(f(x))

and

   3
  f (x) = f(f(f(x)))

and so on.  And we also say that f^0(x) is the identity function x. 
This notation is convenient because it satisfies many of the 
well-known properties of exponents, such as

   n  m        n+m
  f (f (x)) = f   (x).

In this notation, it makes perfect sense to write f^-1(x) for the
inverse function, because then

   -1
  f  (f(x)) = x

is exactly the formula I wrote above, with n = -1 and m = 1.

Finally, we come to the trig functions.  Here, we run into a problem.
You see, when you want to write

      2
  f(x)

where f(x) = sin x, then

       2
  sin x

looks more like

  f(x^2).

So we have two choices.  We could write

  sin(x)^2,

and sometimes you see this (such as in computer math programs), or we
could write

     2
  sin  x,

and sometimes you see this, such as in the Pythagorean trig identity

     2        2
  sin  x + cos  x = 1.

The trouble with this notation is that it means the square of the sin
of x, when it looks more like the iterated

  sin sin x,

so it can be confusing.  It is especially confusing when someone uses
the same notation to write

     -1
  sin  x

to mean the inverse sine of x, because this means the function inverse
(-1 iterations of sine) not the multiplicative inverse (sin x raised
to the -1 power).  It is because of this inconsistency that my
personal preference is always to use "arcsin" to mean the inverse sin,
instead of the -1 exponent.  But not everyone follows this convention.

So the bottom line is that a sine (or cosine, etc.) raised to the -1
power probably means the function inverse and NOT the multiplicative
inverse, while a sine (or cosine, etc.) raised to the power 2 (or any
other positive integer) probably means the number raised to that
power, and NOT the iterated function.  In any case, you can usually
tell by the context.

If you have any questions about this or need more help, please write
back, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/