Is this system invertible?

Question

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$
I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But that does not solve the problem completely. Any help would be appreciated.

Answer 1

Notice that as you integrate over $-\infty<t<\infty$, the right-hand side covers all possible values of $t$. So, the right-hand side will just be constant. Nowhere does changing the value of $t$ in the $y(t)$ on the left-hand side change anything. We could rewrite the equation as: $$y(t)=\int_{-\infty}^{\infty}\frac{x(s)^2}{x(s-1)}ds$$ and it would mean the same thing. Therefore, $y$ is not invertible: it takes on the same value everywhere, so any attempt at an inverse would not be well-defined.