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

In summary, a reflection over the y = -x line can be represented by -R^{-1}(-x), where R is the relation of the original function. This is because every function is a relation and its inverse can be defined as R^{-1}.
  • #1
hb20007
18
0
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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  • #2
Its [itex] f^{-1}(x) [/itex]
Very beautiful!
 
  • #3
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.
 
  • #4
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.
 
  • #5
Understood. My point was that you can't refer to it as f-1(x).
 
  • #6
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.
 
  • #7
Okay, now how about a reflection over y = -x?
 
  • #8
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].
 
  • #9
Yeah, makes sense...
Thanks :biggrin:
 

1. What does it mean for F(-x) to be a reflection over the y-axis of -f(x)?

When F(-x) is a reflection over the y-axis of -f(x), it means that the graph of F(-x) is a mirror image of the graph of -f(x) over the y-axis. This means that any point (x, y) on -f(x) will be reflected to the point (-x, y) on the graph of F(-x).

2. How is a reflection over the y-axis different from a reflection over the x-axis?

A reflection over the y-axis reflects points across the y-axis, while a reflection over the x-axis reflects points across the x-axis. This means that for a reflection over the y-axis, the x-coordinates of the points stay the same, but the y-coordinates change sign. In contrast, for a reflection over the x-axis, the y-coordinates stay the same, but the x-coordinates change sign.

3. Can F(-x) be a reflection over both the x-axis and the y-axis of -f(x)?

Yes, it is possible for F(-x) to be a reflection over both the x-axis and the y-axis of -f(x). This would mean that the graph of F(-x) is a mirror image of the graph of -f(x) over both the x-axis and the y-axis. In this case, any point (x, y) on -f(x) would be reflected to the point (-x, -y) on the graph of F(-x).

4. How can I determine if a function is a reflection over the y-axis?

A function is a reflection over the y-axis if it has the form F(-x) and its graph is a mirror image of the graph of -f(x) over the y-axis. This means that the function must have an even degree and all of its coefficients must be positive.

5. How is a reflection over the y-axis related to the concept of symmetry?

A reflection over the y-axis is a type of symmetry known as y-axis symmetry. This means that the graph of the function is symmetric about the y-axis, and any point on one side of the y-axis has a corresponding point on the other side with the same y-coordinate. Functions with y-axis symmetry are also known as even functions.

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