Re: Domain and range

• To: mathgroup at smc.vnet.net
• Subject: [mg54126] Re: Domain and range
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Fri, 11 Feb 2005 03:33:40 -0500 (EST)
• Organization: The University of Western Australia
• References: <cud82o\$33c\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <cud82o\$33c\$1 at smc.vnet.net>,
"DJ Craig" <spit at djtricities.com> wrote:

> How can I make Mathematica give me either a list or an inequality
> representing all real values of x that will make f[x] either undefined,
> imaginary, or infinite?  For example, if f[x] := (2x^2-18)/(x+3) then x
> != {-3}.  (-3 would give a 0 in the denominator)
> Secondly, how can I find all real numbers that CANNOT be returned by
> f[x] for any real, finite value of x.  In the example, f[x] != {-12}.

f[x_] := (2 x^2 - 18)/(x + 3)

you can find the poles as follows

Solve[Denominator[f[x]] == 0, x]
{{x -> -3}}

This is a removable singularity.

Limit[f[x], x -> -3]
-12

Here is the simplified function:

g[x_] = Simplify[f[x]]
2 (x - 3)

Using Interval arithmetic, we see that -12 is exlcuded:

g[Interval[{-Infinity, -3}]]
Interval[{-Infinity, -12}]

g[Interval[{-3, Infinity}]]
Interval[{-12, Infinity}]

> As a second example, ArcSin[x] only returns a real, finite number when
> -1 <= x <= 1.

Here you can use

Reduce[Element[ArcSin[x], Reals], x]
-1 <= x <= 1

> When it is passed a real, finite number, it can only
> return values where -pi/2 <= ArcSin[x] <= pi/2.

or Interval arithmetic:

ArcSin[Interval[{-1, 1}]]
Interval[{-(Pi/2), Pi/2}]

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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