The pressure PPP of a gas in a specialized containment unit is modeled as a function of its volume vvv by
P(v)=5v+4v−2,v∈R,v≠2 P(v) = \frac{5v + 4}{v - 2}, \quad v \in \mathbb{R}, v \neq 2 P(v)=v−25v+4,v∈R,v=2The volume VVV of the unit varies with time ttt according to the function
V(t)=9−2t2,t∈R,t≥0 V(t) = 9 - 2t^2, \quad t \in \mathbb{R}, t \ge 0 V(t)=9−2t2,t∈R,t≥0Determine the exact time ttt when the pressure reaching the unit is 3, by solving the equation P(V(t))=3P(V(t)) = 3P(V(t))=3.
Find the inverse function P−1(x)P^{-1}(x)P−1(x).
Sketch and label, on the same axes, the curve with equation y=V(x)y = V(x)y=V(x) and the curve with equation y=V−1(x)y = V^{-1}(x)y=V−1(x). Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.
