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Showing Two Functions Are Symmetric About A Line

  1. Mar 1, 2014 #1
    Hello everyone,

    I have the functions [itex]y_1 = \frac{c}{b} + d e^{-bx}[/itex] and[itex]y_2 = \frac{c}{b} - d e^{-bx} [/itex], where [itex]c \in \mathbb{R}[/itex], and [itex]b,d \in \mathbb{R}^+[/itex].
    What I would like to know is how to show that these two functions are symmetric about the line [itex]y = \frac{c}{d}[/itex].

    What I thought was that if y_1 and y_2 are symmetric about the line [itex]y = \frac{c}{d}[/itex], then the distance between y_1 and y, and the distance between y_2 and y, will be the same. That is, [itex]d_1 = \sqrt{(y_1 - y)^2 + (x - x_0)^2}[/itex] and [itex]d_2 = \sqrt{(y_2 - y)^2 + (x - x_0)^2}[/itex], where [itex]d_1 = d_2[/itex].

    Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?
     
    Last edited: Mar 1, 2014
  2. jcsd
  3. Mar 1, 2014 #2

    Ray Vickson

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    Some comments:

    (1) Your distance formula should not involve x, because for each x separately you want to show that the distance between the points (x,c/b) and (x,y_1(x)) is the same as the distance between the points (x,c/b) and (x,y_2(x)). The x drops out of these distance formulas (although, of course, they still contain y_1(x) and y_2(x)). After that, what you say would be correct.

    (2) There is a much easier way.
     
    Last edited: Mar 1, 2014
  4. Mar 1, 2014 #3
    And what might this easier method be, Ray?
     
  5. Mar 1, 2014 #4

    Ray Vickson

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    That is for you to think about; I am not allowed to give solutions, nor would I want to. I can make one suggestion, however: think about what you would get if you drew graphs of the two functions on the same plot.
     
  6. Mar 2, 2014 #5
    I have already drawn the plot of these functions, and that was how I made inference I made, that the distances must be the same. I am not sure what else could be concluded from the plots.
     
  7. Mar 2, 2014 #6
    Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?
     
  8. Mar 2, 2014 #7

    Ray Vickson

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    Well, how would you write it after correcting your erroneous expressions given before?
     
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