Answer by Daniel Huber for Trouble finding inverse of a function

Look at your data:

ListLinePlot[UsvsL, AxesLabel -> {"U", "L"}]

Note that for some Ls there two Us. Therefore the inverse function is not single valued. We therefore create two inverse functions, One with the data from U= 1.01 .. 1.21 and a second one with data from U= 1.21 .. 2.01.

dat1 = Select[UsvsL, #[[1]] <= 1.21 &];dat2 = Select[UsvsL, #[[1]] >= 1.21 &];

Now, how to create the inverse funtions? We already have a table of L[Us]. If we reverse the table entries, we have a table Us[L], but maybe not for the requested L. Toward this aim, we can calculate an interpolating function.

intpol1 = Interpolation[Reverse /@ dat1]intpol2 = Interpolation[Reverse /@ dat2]

With this two functions you may get the Us for given Ls:

Plot[intpol1[l], {l, Min[dat1[[All, 2]]], Max[dat1[[All, 2]]]},  AxesLabel -> {"L", "U"}]Plot[intpol2[l], {l, Min[dat2[[All, 2]]], Max[dat2[[All, 2]]]},  AxesLabel -> {"L", "U"}]