Determining invertibility of weird system

Student here, As an academic exercise, is the system $$y[n] = x[n-1]x[2n]$$ Invertible?

I thought it was since you could find an infinite multiplication series for y[n] that allows you to recover x.

Could someone go over exactly what one to one would mean for a function from X*X->Y? I have only learned one to one functions in context of functions where an input wasn't repeated.

• hint: you can show whether something is not invertible, simply by coming up with an example of where inversion doesn't work. Try a number through which division doesn't help you! Feb 8, 2023 at 19:27

If you consider the following system: $$x[0]=1 \quad x[2]=1 \quad x[k]=0\text{ for } k\neq0,2$$ Then you have the following output: $$y[1]=1 \text{ and }\forall k\neq1\quad y[k]=0$$ By remarking that the same output can be produced by the following system : $$x[0]=-1 \quad x[2]=-1 \quad x[k]=0\text{ for } k\neq0,2$$ you have shown that two different $$x$$ can give you the same $$y$$. This condition indicates that the system is not invertible.