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.

```

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