Inconsistent behavior in ColorFunctionScaling of BarLegend in v10.0.1

Question

I find the ColorFunction in BarLegend behaves very weirdly:

1. Force rescaling from 0 to 1

   For example:

   cf1[z_] := ColorData[{"Temperature", {0, 1}}][z];
   BarLegend[{cf1[#] &, {0, .5}}]
   

   gives
   

2. Rescaling changes within MatrixPlot

   For example:

   MatrixPlot[RandomReal[{1, 4}, {10, 10}],
   PlotLegends -> BarLegend[{"Temperature", {1, 4}}],
   ColorFunction -> (ColorData[{"Temperature", {1, 4}}][#] &),
   ColorFunctionScaling -> False, DataReversed -> True]
   

   gives
   

3. However, generating BarLegend alone as follows:

   BarLegend[{"Temperature", {1, 4}}]
   

   gives
   

Obviously, ColorFunctionScaling in BarLegend has been set to False accordingly as in MatrixPlot, which is unexpected since I have no control of the scaling inside BarLegend.

Questions:

Is the auto-rescaling a bug?
Or, how can I "utilize" the feature?

Answer 1

MatrixPlot[RandomReal[{1, 4}, {10, 10}],
 PlotLegends ->  BarLegend[{ColorData[{"Temperature", {1, 4}}], {1, 4}}],
 ColorFunction -> (ColorData[{"Temperature", {1, 4}}]),
 ColorFunctionScaling -> False, DataReversed -> True]

Version 9.0.1.0:

Version 10.0.1.0 (Wolfram Programming Cloud):

Update: PlotLegends -> BarLegend[{ColorData[{"Temperature", {1, 4}}], {1, 4}}, 10] does not work as expected to add contours in the bar legend; it rescales the tick labels to {0,1}. A work-around is to use Legended:

Legended[MatrixPlot[RandomReal[{1, 4}, {10, 10}],
  ColorFunction -> (ColorData[{"Temperature", {1, 4}}]),
  ColorFunctionScaling -> False, DataReversed -> True], 
 BarLegend[{ColorData[{"Temperature", {1, 4}}], {1, 4}}, 10]]

Update 2: version 11.3 (Wolfram Programming Cloud)

$Version

"11.3.0 for Linux x86 (64-bit) (February 20, 2018)"

 MatrixPlot[RandomReal[{1, 4}, {10, 10}],
  PlotLegends ->  BarLegend[{ColorData[{"Temperature", {1, 4}}], {1, 4}}],
  ColorFunction -> (ColorData[{"Temperature", {1, 4}}]),
  ColorFunctionScaling -> False, DataReversed -> True]

Use BarLegend[{ColorData[{"Temperature", {1, 4}}], {1, 4}}, 10] to get