myfun[a_, l1_, l2_] =
 Piecewise[{{a^2, a < l1},
            {a - l1 + l1^2, a >= l1 && a < l2},
            {l2 - l1 + l1^2 + Log[a - l2 + 1], a >= l2}}]

Plot[Evaluate[myfun[a, l1, l2] /. {l1 -> 1.5, l2 -> 3.2}], {a, 0, 5}, PlotRange -> All]

myInvFun[l1_, l2_] := x /. FindRoot[myfun[x, l1, l2] == 2.0, {l, 0}]

Plot[Evaluate[myInvFun[l1, l2] /. {l2 -> 3.5}], {l1, 0, 3.5}]