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I want an operator that takes as input a function of a real variable e.g. f(x) and returns as output the answer f(-x). How does one define such an operator in Mathematica?

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    $\begingroup$ f[x] /. h_[a_]:>h[-a] but I guess the best solution may be background dependent. So what it the big picture here? What have you tried? $\endgroup$ – Kuba Sep 12 '17 at 10:08
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    $\begingroup$ Try f @* Minus. $\endgroup$ – J. M.'s technical difficulties Sep 12 '17 at 16:23
  • $\begingroup$ Thanks for all answers. I am looking for Pure functions only. Some of these will work for me... $\endgroup$ – Quasar Supernova Sep 13 '17 at 12:04
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You can define such a operator like so.

yAxisReflect[f_] := (f[-#] &)

where f is a symbol naming a function or a pure function.

With this definition, you can do things like

Plot[{Sin[x], yAxisReflect[Sin][x]}, {x, -π, π}]

sine

Plot[{(1 + x)^2, yAxisReflect[(1 + #)^2 &][x]}, {x, -2, 2}]

square

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One approach is to define a "negative" function directly:

g[x_] := f[-x]

So however f[x] is defined, g[x] gives f[-x]. But really, why do this? Wouldn't it be simpler and clearer just to use f[-x] directly whenever needed?

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ClearAll[reflectionF1, reflectionF2]
reflectionF1[f_] := Compose[f, Minus, #] &;
reflectionF2[f_] := Composition[f, Minus] (* f @* Minus  in V 10.0+ as suggested by JM *)

Using m_goldberg's example setup:

Row[{Plot[{Sin[x], reflectionF1[Sin][x]}, {x, -\[Pi], \[Pi]}, 
   PlotLabel -> Style["reflectionF1", 16, "Panel"], ImageSize -> 400, PlotStyle -> Thick], 
  Plot[{Sin[x], reflectionF2[Sin][x]}, {x, -\[Pi], \[Pi]},
   PlotLabel -> Style["reflectionF2", 16, "Panel"], ImageSize -> 400, PlotStyle -> Thick]}]

enter image description here

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