A signal processing system has a gain factor GGG defined by the function
G(v)=v+8v−1 G(v) = \frac{\sqrt{v+8}}{v - 1} G(v)=v−1v+8where GGG has its maximum possible domain in terms of the input voltage vvv.
Determine the domain of GGG. Give your answer using set notation.
An engineer calculates G(0)≈−2.83G(0) \approx -2.83G(0)≈−2.83 and G(2)≈3.16G(2) \approx 3.16G(2)≈3.16. They argue that because there is a change of sign, there must be a voltage vvv in the interval 0<v<20 < v < 20<v<2 such that the gain factor G(v)=0G(v) = 0G(v)=0. Explain the error in the engineer's argument.
By considering any turning points of GGG, determine whether GGG has an inverse function. Fully justify your answer.
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.