Re: finding inverses of functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg126441] Re: finding inverses of functions*From*: "djmpark" <djmpark at comcast.net>*Date*: Thu, 10 May 2012 05:00:52 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <4767427.124602.1336550815297.JavaMail.root@m06>

An interesting problem. I'm going to take your "e" to be Exp. Let's define the y function: Clear[x, y]; y[x_] := 3 x^3 + 2 E^(2 x); The function looks to be monotonically increasing. Plot[y[x], {x, -2, 2}] And we can verify it. ForAll[x, D[y[x], x] > 0] Resolve[%, Reals] ... True We will solve a differential equation and need an initial condition: y[0] 2 Write the inverse slope in terms of x[y] and solve the differential equation. Clear[x] x'[y] == 1/(D[y[x], x] /. x -> x[y]); xsol = DSolve[%, x, y][[1, 1]] x -> Function[{y}, InverseFunction[2 E^(2 #1) + 3 #1^3 &][y + C[1]]] Solve the initial condition for C[1]. x[2] == 0 /. xsol; Solve[%, C[1]][[1, 1]] C[1] -> 0 We can now define the x function. It is in terms of an InverseFunction but easily evaluable. x[y_] = x[y] /. (xsol /. C[1] -> 0) InverseFunction[2 E^(2 #1) + 3 #1^3 &][y] We can check numerically that the function is inverse, at least for small or exact values of x. x[y[x]] == x; % /. x -> 10 True (I wish I knew a method to show this symbolically.) We can plot the y function, inverse x function and their composition to check, at least numerically, the proper relation. Show[ {Plot[y[x], {x, -2, 2}, PlotStyle -> Black], Plot[x[y], {y, -20, 20}, PlotStyle -> Blue], Plot[y[x[z]], {z, -10, 10}, PlotStyle -> Red]}, AspectRatio -> Automatic, PlotRange -> 10, Axes -> False, Frame -> True] David Park djmpark at comcast.net http://home.comcast.net/~djmpark/index.html From: John Accardi [mailto:accardi at accardi.com] Hello, I am trying to get Mathematica to find the inverse of: y = 3x^3 + 2e^(2x) (which I know is invertible) InverseFunction only seems to give inverses of built-ins, like Sine. I tried: Solve[ y == 3x^3 + 2e^(2x), x ] but get a message that Solve does not have methods suitable. (Solve works for simpler functions, however.) Any ideas? Thanks.