Gas Storage Tank Pressure Thresholds

In industrial mechanical and chemical plant engineering, pressurized gas containment vessels operate under automated digital monitoring to prevent explosive rupture or structural implosion. Under high-temperature thermodynamic regimes, non-ideal gases deviate from linear equations of state, exhibiting asymptotic pressure surges when thermal and volumetric inputs approach acoustic resonance or critical condensation limits.

A UNLV mechanical testing lab monitors the internal pressure $P$ (in megapascals, $\text{MPa}$) of an experimental carbon dioxide storage vessel using the rational multivariable model:

where $x$ represents normalized core gas temperature (in hundreds of Kelvin, $x > 0$) and $y$ represents a dimensionless chamber volume expansion ratio ($y > 0$).

During a dynamic test cycle, a control valve malfunction attempts to set the operational parameters to $(x, y) = (2, 2)$.

1. State the algebraic domain condition for $g(x, y)$ and write the mathematical domain $D$ in set-builder notation. Describe the geometric curves in the $xy$-plane that must be excluded from the operational domain.
2. Determine the mathematical range $R$ of the pressure function across its unrestricted domain and across the physical positive quadrant ($x > 0, y > 0$).
3. Evaluate the pressure function at the target settings $(x, y) = (2, 2)$. Explain what happens as operating conditions approach this point, and discuss how automated safety control logic must respond.