A signal processing unit models the gain G G\,G as a function of input voltage v v\,v according to the rule
G(v)=5−3v2v+1,v∈R,v≠−0.5 G(v) = \frac{5 - 3v}{2v + 1}, \quad v \in \mathbb{R}, v \neq -0.5 G(v)=2v+15−3v,v∈R,v=−0.5The input voltage v v\,v is determined by the time t t\,t in seconds via the function
V(t)=4−2t,t∈R V(t) = 4 - 2t, \quad t \in \mathbb{R} V(t)=4−2t,t∈REvaluate the composite gain GV(−1.5)GV(-1.5)GV(−1.5).
Obtain an expression for the inverse gain function G−1(x)G^{-1}(x)G−1(x).
Determine the values of k k\,k that satisfy the equation G(1k)=V(k+1)\displaystyle G\left(\frac{1}{k}\right) = V(k + 1)G(k1)=V(k+1).
