RE: Simple question about inverse of a function
- To: mathgroup at smc.vnet.net
- Subject: [mg122537] RE: [mg122502] Simple question about inverse of a function
- From: "David Park" <djmpark at comcast.net>
- Date: Mon, 31 Oct 2011 06:51:51 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <30343966.37232.1319964520601.JavaMail.root@m06>
I like this kind of question because it reminds us of Mathematica capabilities that we might be less familiar with. I can't believe this hasn't been enhanced since Version 2, but perhaps indirectly through Solve. Since theta is a parameter let's put it in as a SubValue. Clear[f, g] conditions = 0 <= t <= 1 && 1 <= theta <= Infinity; f[theta_][t_] := (1 - t)^theta g[theta_] = Assuming[conditions, InverseFunction[f[theta]]] 1 - #1^(1/theta) & and a warning message that I believe can be ignored. Testing: Simplify[g[theta][f[theta][t]], conditions] t The following draws the function, the inverse by interchanging x and y, the calculated inverse on top of it, and the inverse operating on the function in blue. << Presentations` With[{theta = 5}, Draw2D[ {Draw[f[theta][t], {t, 0, 1}, PlotRange -> All], {Opacity[0.2, Black], AbsoluteThickness[7], ParametricDraw[{f[theta][t], t}, {t, 0, 1}]}, {Red, Draw[g[theta][t], {t, 0, 1}, PlotRange -> All]}, {Blue, Draw[g[theta][f[theta][t]], {t, 0, 1}]}}, AspectRatio -> 1, PlotRange -> Automatic, Frame -> True, FrameLabel -> {t, None, None, None}, ImageSize -> 200] ] David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: Mikael [mailto:mikaen.anderson.1969 at gmail.com] I have a simple question on how to calculate the inverse of a a function. This is the function I define: f[t_] := (1 - t)^theta To calculate the inverse I write: Assuming[t >= 0 && t <= 1 && theta >= 1 && theta < Infinity, { InverseFunction[f[t]]}] but the answer I get is {InverseFunction[(1 - t)^theta]}. Now I know I can do this: In[11]:= Solve[f[g[x]]==x,g[x]] Out[11]= {{g[x]->1-x^(1/theta)}} but I wonder what is the correct way of specifying assumptions on t and theta to make the InverseFunction work. Thanks.