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).
471 exam-style questions on Edexcel A Level Maths Functions and Graphs, covering 2.1 The Modulus Function, 2.2 Functions and Mappings, 2.3 Composite Functions, 2.4 Inverse Functions, 2.5 y=|f(x)| and y=f(|x|), 2.6 Combining Transformations, and 2.7 Solving Modulus Problems. Each one has a worked solution and a mark scheme showing where the marks go.