I found this forum after experiencing the same problems with our PX-410R which had stopped working and exhibiting the same symptoms: not reacting to the Transpose/Function button, the three Mixer, Synth and Effect buttons right underneath are not reacting either, and the note sound and corresponding note on the display were also incorrect relative to what key was actually pressed.
As the Transpose / Function button was not responding, I was not able to access the Initialize Settings as instructed on page E-69 of the user guide. (A good argument for why a separate re-set button on the rear panel would be desirable.) With no solution confirmed in this thread, I even started to look at replacement options on-line, and baulking at the cost of an upgrade.
The next morning, I dug up a 12V 2.0 Amp power supply, and decided to give it ago as described in the above thread. When I turned the power on, the display screen "Flashed" momentarily, and then nothing. The alternate power supply was a generic supply that I had for another device, and the fit into the socket was not as snug as the original power supply, so I decided to try it again while pressing the plug in to ensure a good connection. - Nothing at all this time. Not even a screen flash.
Thinking that it was well and truly dead, I then plugged the original 12 V 1.5 Amp Casio power supply back in and presto, the screen lit up, and everything restored back to normal operations!!!!
Thank you to all who have contributed to this forum. I think that what Jokeyman123 mentioned about the replacement of the battery may be worth considering. Can anyone advise on how to access the battery? - I am assuming that there is the equivalent of a CMOS battery inside to assist with the retention of stored data when not in use.
