Reflecting functions over the y-axis, x-axis, and the line y=x

[hb20007]

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?

 

[ShayanJ]

Its $f^{-1}(x)$
Very beautiful!

 

[Mark44]

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.

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.

 

[ShayanJ]

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.

 

[Mark44]

Understood. My point was that you can't refer to it as f^-1(x).

 

[R136a1]

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.

 

[hb20007]

Okay, now how about a reflection over y = -x?

 

[ShayanJ]

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...$-R^{-1}(-x)$.

 

[hb20007]

Yeah, makes sense...
Thanks