Showing Two Functions Are Symmetric About A Line

[Bashyboy]

Hello everyone,

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

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

Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?

 

[Ray Vickson]

Hello everyone,

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

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

Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?

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.

 

[Bashyboy]

And what might this easier method be, Ray?

 

[Ray Vickson]

And what might this easier method be, Ray?

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.

 

[Bashyboy]

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.

 

[Bashyboy]

Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?

 

[Ray Vickson]

Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?

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