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Re: algebraic functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg29109] Re: [mg29090] algebraic functions
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Wed, 30 May 2001 05:50:24 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

 I guess that by "solving a function" you are referring to finding inverse
images of values in the range of your function?
Also, presumably, by algebraic functions you mean  rational functions
(including also polynomials). (At least that is the usual meaning.). In such
cases you can find "the range" as in the following example:

In[1]:=
f[x_] := 5/(x - 1)^2
In[2]:=
f[Interval[{-Infinity,Infinity}]]
Out[2]=
Interval[{0, Infinity}]

There is a small problem with this answer. I suppose you do not wish to
consider 0 as being  in the range of the function  but Mathematica includes
it (Interval[{a,b}] means a closed interval). So you have to adjust for it
yourself.

Also, presumably by domain you mean the set of real values where the formula
does not become infinite. For rational functions you can find the points
"not in the domain" by

In[3]:=
Union[x/.Solve[Denominator[f[x]]==0,x]]
Out[3]=
{1}

Perhaps I should also add that from the strict mathematical point of view
your question does not make much sense. The domain of a function cannot be
"deduced" from "the formula", it has to be given as part of the definition
of the function. For example, exactly the same formula as above can be
considered as defining a continuous function in the complex plane minus the
complex number 1. It's range will then consist of the entire complex plane
except 0. Or it could be the function defined on the Riemann sphere (in
other words, including also 1 and  ComplexInfinity). It's range will then be
the entire Riemann sphere and the function will be a holomorphic mapping. Or
it could be considered as a function defined only on some subset of the real
numbers, e,g, an interval and so on.
Presumably what you learned to call "the domain" is what, in this case,
should be described the largest subset of the real numbers on which the
formula defines a continuous real valued function. Sometimes people seem t
mean by "the domain"  the largest set for which a formula "makes sense", but
all the definitions are just too loose to be useful. Just ask yourself the
question: what is the domain of the function f[n_]:=n! or f[x_]:= 1/(x -
Sqrt[3])! 


-- 
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/~andrzej/


on 01.5.29 3:57 PM, Steven Shippee at slshippee at home.com wrote:

> 
> I have version 4.1 of Mathematica.
> 
> I have read the documentation, and found that functions could be solved
> by typing f[x_] := 3x - 7 and solving for multiple instances of x by
> f[1] = with the number in brackets being the variable for x.
> 
> But I can not find how to have mathematica compute "domain" and "range"
> of algebraic functions . is this possible??
> 
> 
> Thank you very much in advance,
> 
> Steven Shippee
> 
> 1-360-902-5817
> 
> mailto:shippee at jcs.mil
> 
> 
> 
> 



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