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

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The discussion clarifies that f(-x) represents a reflection over the y-axis, while -f(x) indicates a reflection over the x-axis. It explores how to represent a reflection over the line y = x, which is denoted as f^{-1}(x), but notes that this only applies if the function is one-to-one. For functions like y = f(x) = x^2, which lack an inverse, the reflection can still be defined as a relation rather than a function. The conversation also touches on reflections over the line y = -x, concluding with the expression -R^{-1}(-x) for that transformation. Understanding these reflections is essential for grasping function behavior in mathematics.
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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?
 
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Its f^{-1}(x)
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 f^{-1}(x)
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 y=x^2 we have x=\pm \sqrt{y}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 (x_0,y_0)\rightarrow(-y_0,-x_0).
It think it should be -f^{-1}(-x)...ohh...sorry...-R^{-1}(-x).
 
Yeah, makes sense...
Thanks :biggrin:
 

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