Inverting a function in a certain region

Question

InverseFunction works well for globally invertible functions, like

f = 2*# + 2 &;
InverseFunction[f]

1/2 (-2 + #1) &

However, how would one for example find the inverse of $f(x)=x^2$ around $(x,f(x))=(1,1)$? My Mathematica (8.0.4) gives me a warning about multivalued inverses in this case, as well as the result $-\sqrt{x}$, which is not the branch that I'm looking for:

f = #^2 &
InverseFunction[f]

#1^2 &

InverseFunction::ifun: 
  Inverse functions are being used. Values may be lost for multivalued inverses. >>

-Sqrt[#1] &

So: How can I specify the neighbourhood a function should be inverted on?

Answer 1

Something like this is helpful :

InverseFunction[ConditionalExpression[#1^2, 2 > #1 > 0] &]

yields

ConditionalExpression[Sqrt[#1], 0 <= #1 <= 4] &

Then you can use it as an ordinary function, e.g. :

Integrate[%[x], {x, 1/2, 3/2}]

1/6 (-Sqrt[2] + 3 Sqrt[6]) 

or

D[ConditionalExpression[Sqrt[#1], 1/4 <= #1 <= 9/4] &[x], x]

ConditionalExpression[1/(2 Sqrt[x]), 1/4 <= x <= 9/4]

Plot[ConditionalExpression[Sqrt[#1], 1/4 <= #1 <= 9/4] &[x],
                         {x, 1/4, 9/4}, AxesOrigin -> {0, 0}]

Answer 2

You can also use Solve (or Reduce) to obtain the inverse function and pick the branch that matches your neighbourhood point. A simple approach to obtain $x(y)$ given $y(x)$ and $\{x_0, y (x_0)\}$ is the following:

inverseFunc[func_, x_, y_, {a_, b_}] := Module[{f, sols},
    Pick[#, # /. {Rule -> Equal, x -> a, f -> b} // Flatten] &@
      Solve[f == func, {x}] /. f -> y
]

For your example of $f(x)=x^2$:

inverseFunc[x^2, x, y, {1, 1}]
Out[1]= {{x -> Sqrt[y]}}

Answer 3

Being a nonexpert, this is how I invert any kind of weird functions:

f[x_]:=x^2;
invf=Interpolation[Table[{f[x],x},{x,3,6,0.1}]]

InterpolatingFunction[{{9.,36.}},<>]

invf[25]

5.