You work in the company „HHN Mechatronics & Robotics“ and are supposed to generate a bipolar signal ($-10 ~\rm V ... + 10 ~\rm V$) from a unipolar signal of a digital-to-analog converter ($0 ... 5 ~\rm V$) 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 $U_\rm O$ and $U_\rm I$ as $U_{\rm O}(U_{\rm I})$ by superposition.
3. The circuit still has the problem that for a positive positive half-wave the output is still negative. Which additional circuit must be provided so that this problem can be solved?