It's actually quite easy to be able to tell. You just have to understand what the inverter does.
All the inverter does is provide AC power to the bulb that lights the screen. That's it. It inverts the DC power from the battery or power adapter, to AC power that allows the cold cathode fluorescent bulb to work.... since that bulb needs several thousand volts to ignite.
So, if the inverter wasn't working, the screen would not light. You would still see the desktop and mouse and icons dimly, but it would be exactly as it sounds... no light.
The issues you describe are with the LCD panel itself. There is a small (2%) chance it is with the integrated video chipset, but you can eliminate that possibility by connecting an external desktop monitor to the VGA port on the laptop. If you see the same problem on the external screen, you know it is an issue with the motherboard/integrated video. However, I'm 98% sure that your issue is with the LCD panel, and will require a replacement of the screen.