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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}.
Some ideas. For your function,
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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