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