Sketching the Graph of $x=f(y)$

Question

In order to sketch a graph of $x=f(y)$ when we know the graph of $y=f(x)$ We can exchange the axes of coordinates (and sketch the same graph of $y=f(x)$ on the other axis) I want use this method to sketch graph of $x=y^3$ (we know it is the grapgh of $y=\sqrt[3]{x}$). I get this graph:

I used the method and I rotated $x$ and $y$ axis $90^\circ$ clockwise in my head and draw the graph ( I showed where $x$ and $y$ axis are positive or negative after rotation ). But this graph is wrong and I realized that $x^+ , x^-$ are misplaced and should be exchanged.

Can you explain what is wrong with the argument that We should visualize rotating x and y axes and then sketch the graph normally instead of considering $x^+$ as the upper ray.

Thank you.

Answer 1

To see what is wrong with your approach, look at the following (admittedly not very good quickly put together in paint) diagram:

You want to draw ${x}$ as a function of $y$, then do some form of rotation to get back to the standard "${y}$ as a function of ${x}$" graph - rotate each of these pictures by ${360}$ degrees in your head. You will see that at no point during the rotation do you get the other. And so there does not exist a rotation that gets you from one state to the other. As others have pointed out - you must reflect. We can get close though - a rotation by ${270}$ degrees clockwise of the "${x}$ as a function of ${y}$" diagram would give you

But this isn't the same as the "${y}$ as a function of ${x}$" graph - it's actually "${y}$ as a function of ${-x}$". Hope that helps

Edit: @b00n heT has made a lovely comment I'd like to just point out - the problem is that any rotation on the "${y}$ as a function of ${x}$" graph will preserve the fact that the ${x}$ axis remains to the right of the ${y}$ axis. In the "${x}$ as a function of ${y}$" graph - the $x$ axis is to the left of the $y$ axis - and so it is clear no rotation on one will give you the other.

Answer 2

What you want is to reflect about the line $y=x$.

Answer 3

after studying answers I finally conclude :

We Have graph of $y=f(x)$ and want to sketch the graph of $x=f(y)$. As we interchanged $x$ and $y$ in the equations we should interchange name of the $x,y$ axis too. So in the diagram of the question, $x^+$ should be at the up of $y$ axis and $x^-$ should be on the down of the $y$ axis.

So if I insist on sketching the graph by the method I mentioned in the question, the final step is reflecting the graph about the $x$ axis to correct the sign of $x$.