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

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

Mark44
Mentor
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
Gold Member
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
Mentor
Understood. My point was that you can't refer to it as f-1(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.

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?

ShayanJ
Gold Member
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 