F(-x) is a reflection over the y axis -f(x)

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Discussion Overview

The discussion revolves around the concept of reflections of functions across various axes and lines, specifically focusing on reflections over the y-axis, x-axis, and the line y=x. Participants explore how to represent these reflections mathematically and the implications when functions do not have inverses.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants state that f(-x) represents a reflection over the y-axis, while -f(x) represents a reflection over the x-axis.
  • There is a proposal that a reflection over the line y=x can be represented as f^{-1}(x), although this is challenged by the existence of functions that do not have inverses.
  • One participant points out that for functions like y = f(x) = x², which are not one-to-one, there is no inverse function, but a relation can be defined.
  • Another participant emphasizes that while a function may not have an inverse, the inverse relation can still be plotted.
  • A later post introduces the concept of reflecting over the line y=-x, suggesting that this can be represented as -R^{-1}(-x).

Areas of Agreement / Disagreement

Participants express differing views on the representation of reflections and the implications of functions lacking inverses. There is no consensus on how to handle cases where a function does not have an inverse, and the discussion remains unresolved regarding the best mathematical representation for these reflections.

Contextual Notes

Limitations include the assumption that all functions are one-to-one when discussing inverses, and the discussion does not resolve the mathematical steps involved in defining inverse relations versus inverse functions.

hb20007
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f(-x) is a reflection over the y axis
-f(x) is a reflection over the x axis

Now, how do we represent a reflection over y=x?
 
Last edited:
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Its [itex]f^{-1}(x)[/itex]
Very beautiful!
 
hb20007 said:
f(-x) is a reflection over the y axis
-f(x) is a reflection over the x axis

Now, how do we represent a reflection over y=x?

If (x, y) is a point on the graph of f, (y, x) will be the reflection of that point across the line y = x.

Shyan said:
Its [itex]f^{-1}(x)[/itex]
Very beautiful!
What if f doesn't have an inverse? For example, y = f(x) = x2. This function is not one-to-one, so doesn't have an inverse.
 
Mark44 said:
If (x, y) is a point on the graph of f, (y, x) will be the reflection of that point across the line y = x.What if f doesn't have an inverse? For example, y = f(x) = x2. This function is not one-to-one, so doesn't have an inverse.

If a function is not one to one,then there is no function that is its inverse.But there is of course a relation which is the function's inverse.And that relation can be ploted.For [itex]y=x^2[/itex] we have [itex]x=\pm \sqrt{y}[/itex]which is a two-valued relation between x and y.
 
Understood. My point was that you can't refer to it as f-1(x).
 
Mark44 said:
If (x, y) is a point on the graph of f, (y, x) will be the reflection of that point across the line y = x.


What if f doesn't have an inverse? For example, y = f(x) = x2. This function is not one-to-one, so doesn't have an inverse.

Every function is a relation. If ##R## is a relation, then ##R^{-1}## is a well-defined relation.
 
Okay, now how about a reflection over y = -x?
 
Let's see...a reflection over line y=-x means [itex](x_0,y_0)\rightarrow(-y_0,-x_0)[/itex].
It think it should be [itex]-f^{-1}(-x)[/itex]...ohh...sorry...[itex]-R^{-1}(-x)[/itex].
 
Yeah, makes sense...
Thanks :biggrin:
 

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