You work in the company „HHN Mechatronics & Robotics“ and are supposed to generate a bipolar signal ($-10V ... + 10V$) from a unipolar signal of a digital-to-analog converter ($0...5V$) in a project. A colleague recommended the circuit shown on the right.

1. First, analyze what change is made by pressing the switch $S$. How does the output signal change?
2. Try to determine mathematically the relationship of $U_O$ and $U_I$ as $U_O(U_I)$ by superposition.
3. The circuit still has the problem that for an positive positive half-wave the output is still negative. Which additional circuit must be provided so that this problem can be solved?