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.
181 exam-style questions on OCR (MEI) A Level Maths 1.3 Functions, covering 1.3.1 Operations on polynomials, 1.3.2 Factor theorem, 1.3.3 Definition of a function (A-level only), 1.3.4 Composite functions (A-level only), 1.3.5 Inverse functions and their graphs (A-level only), 1.3.6 The modulus function (A-level only), 1.3.7 Inequalities with a modulus sign (A-level only), and 1.3.8 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.