There are several functions of two or three variables that I am working with. For this question I have made a small set showing the resistivity, $\rho$, in n$\Omega$m, of copper as a function of its purity ($RRR$ value) and its temperature, $T$, in Kelvin.
RRR\Temp [K] 20 90 160 230 300 60 0.27 3.1 8.15 12.9 17.5 300 0.06 2.81 7.84 12.6 17.2 500 0.04 2.78 7.81 12.5 17.2 1000 0.02 2.76 7.79 12.5 17.2
So this 'function' has the form
$\rho (T, RRR)$
Now, I do know how to make an interpolating function for two variables in my language of choice but I don't know how to change the variables, say to
$RRR (T, \rho)$
This would be very beneficial as the purity of the sample is only known approximately while the temperature and resistivity are known to great accuracy.
I have seen how to do this where an analytic representation is know by using a transform of independent variables. But when I thought about how I apply that to this to this problem I ran into a wall.
EDIT: To clarify what I mean when I say that the purity is known approximately: a particular sample may claim that its RRR is 100. If I check a sample of 6 AWG (1.33e-5 $m^2$), length 0.1 m at 293 K the calculation says I should see a resistance of 127.07 $\mu\Omega$. There will be a difference from the measured value but I should be able to distinguish the source of error, i.e. incorrect purity, instrumentation error, etc... Taking a data point at several temperatures and converting resistance back in to resistivity will be a good plan but this would necessitate the use of a function of the form:
As to whether it is invertible, I think it is. I will always know the temperature. See contour plot below.