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.
Practise AQA A Level Maths 1.1 OT1: Mathematical argument, language and proof with exam-style questions for A Level Maths. 8 questions covering 1.1.1 Constructing mathematical arguments, 1.1.2 Mathematical language and syntax, 1.1.3 Set theory language and symbols, 1.1.4 Definition, domain and range of functions, and 1.1.5 Comprehending and critiquing arguments, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
