The reflection of f(x) in the vertical line x=k is given by f(2k-x)

reflect_horizontally.txt

Produce a new function which is a horizontal reflection of the function f{x} in the vertical line x=k.

There are three steps: (1) horizontally translate the function in such a way that the reflection can be applied across the 
Y-axis (i.e. all points on the translated function's curve should be the same distance from the Y-axis as they originally
were from the line x=k); (2) perform a reflection across the Y-axis; (3) reverse the translation made in step 1.

1. Consider the translation necessary to make the line x=k coincide with the Y-axis. Apply this translation to f{x} 
   producing f₁{x}.

   f₁{x} = f{x+k}
  
2. Apply a reflection across the Y-axis to f₁{x} producing f₂{x}.

   f₂{x} = f₁{-x}
         = f{k-x}
  
3. Consider the opposite translation to that performed in step 1. Apply this translation to f₂{x} producing f₃{x}.

   f₃{x} = f₂{x-k}
         = f{k-{x-k}}
         = f{2k-x}
         
The horizontal reflection of the function f{x} in the vertical line x=k is given by f{2k-x}.

Example.

Let f{x}=e^(3-x) and let r{x} be the reflection of f{x} in the line x=-4. Find r{x} in terms of x.
  
r{x} = f{2k-x}, where k=-4
     = f{-8-x}
     = e^(3-(-8-x))
     = e^(11+x)