To Work with Barchart or DiscretePlot?

This is a bit easier to achieve with DiscretePlot instead of BarChart.

With

f[x_] := 3 - 1/2 x

Then

Show[
 Plot[f[x], {x, 0, 14}],
 DiscretePlot[f[x], {x, Range[2, 12, 2]}, 
  PlotMarkers -> "Point", 
  ExtentSize -> Right, 
  PlotStyle -> Gray],
 Epilog -> {
   Inset[Row[{Inactivate[f[x]], "=", f[x]}, Spacer[1]], 
    Scaled[{.2, .2}],
    FormatType -> TraditionalForm,
    BaseStyle -> {FontSize -> Scaled[.04]}]
   }
 ]

Hope this helps.

---

Update

Taking cues from this answer (19542) and applying to a custom ExtentElementFunction can be created.

ClearAll[fillRandomDots];
fillRandomDots[{{xmin_, xmax_}, {ymin_, ymax_}}, ___] :=
 Module[{rect = Rectangle[{xmin, ymin}, {xmax, ymax}], dots, texture},
  If[ymax - ymin > 0,
   dots = RandomPoint[rect, Area@rect 600];
   texture =
    Rasterize@
     Graphics[{Opacity[.1, Gray], rect, 
       Opacity[.9, Lighter[Black, .3]], Disk[#, .05] & /@ dots},
      PlotRange -> {{xmin, xmax}, {ymin, ymax}},
      PlotRangePadding -> None,
      ImagePadding -> None
      ];
   ,
   texture = Graphics@{}
   ];
  {
   EdgeForm[{Thin, Black}],
   Texture@texture,
   Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}, {xmin, ymax}},
    VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]
   }
  ]

Then

Show[
 DiscretePlot[f[x], {x, Range[2, 12, 2]},
  PlotMarkers -> "Point",
  ExtentSize -> Right,
  ExtentElementFunction -> fillRandomDots],
 Plot[f[x], {x, 0, 14},
  PlotTheme -> "Monochrome"],
 PlotRange -> {{0, 14}, {-4, 3}},
 AxesOrigin -> {0, 0},
 Epilog -> {
   Inset[Row[{Inactivate[f[x]], "=", f[x]}, Spacer[1]], 
    Scaled[{.2, .2}],
    FormatType -> TraditionalForm,
    BaseStyle -> {FontSize -> Scaled[.04]}]
   }
 ]

Cues can be taken from this answer (19542) to change the sides the axis ticks and labels appear.

You can also use RectangleChart with a custom ChartElementFunction:

ClearAll[ceF, rectChart]
ceF[func_][{{x0_, x1_}, {y0_, y1_}},  ___] := 
  Dynamic@{If[CurrentValue["Color"]=== White, {},
  {Texture[Rasterize[RandomImage[1, {50, 50}]]],
    Polygon[#, VertexTextureCoordinates -> #]&@{{x0, y0}, {x0, y1}, {x1, y1}, {x1, y0}}}], 
    Text[Style[Floor@x0, If[Floor@x0 == 0, White, Black], "Panel", 12], {x0, 0}, 
    Switch[Sign[func /@ {x0 - .01 Mean[{x0, x1}], x0 + .01 Mean[{x0, x1}]}], 
      {1, 1} | {1, -1} | {0, -1}, {0, 1}, {-1, -1} | {-1, 1} | {0, 1}, {0, -1}]]};

rectChart[fun_, from_, to_, o : OptionsPattern[]] := 
  RectangleChart[ArrayPad[Table[{from, fun[x]}, {x, from, to - 1, from}], {{1}}, 
     Style[{from, 1}, White]] /. x : {_, 0} :> Style[x + {0, 1}, White],
   BarSpacing -> 0, o, 
   Epilog -> (Plot[fun[x], {x, 0, to + from}, PlotStyle -> GrayLevel[.1]][[1]]), 
   PlotRange -> {fun[0], fun[to]}, ChartElementFunction -> ceF[fun]];

Examples:

f[x_] := 3 - x/2;
rectChart[f, 2, 14, ChartLegends -> 
  Placed[TraditionalForm[Style[HoldForm@f[x] == f[x], 16, "Panel"]], {.2, .3}]]

f[x_] := 3 - x/2;
rectChart[f, 1, 14, ChartLegends -> 
  Placed[TraditionalForm[Style[HoldForm@f[x] == f[x], 16, "Panel"]], {.2, .3}]]

rectChart[# Pi Cos[# Pi/4]/4 &, 2, 21, PlotRange -> {-20, 20}, 
  ChartLegends -> Placed[TraditionalForm[
    Style[HoldForm@f[x] == x Pi Cos[x Pi/4]/4, 16, "Panel"]], {.3, .9}]]