The cooling efficiency, EEE, of a high-performance server rack is modelled by the function
E(v)=5v−2v−4,v>4 E(v) = \frac{5v - 2}{v - 4}, \quad v > 4 E(v)=v−45v−2,v>4where v v\,v is the coolant flow rate in litres per minute.
Use differentiation to show that E E\,E is a decreasing function on its domain.
Find an expression for E−1(v)E^{-1}(v)E−1(v) and state its domain.
Show that the composite function EE(v)EE(v)EE(v) can be written in the form av+bv+c\displaystyle \frac{av + b}{v + c}v+cav+b, where a,b, a, b,\,a,b, and c c\,c are integers to be found.
Deduce the range of EEEEEE.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.