The light intensity III (in lumens) of a specialized lamp at a distance x x\,x meters is modeled by the function
I(x)=40x+3,x>0,x∈R I(x) = \frac{40}{x + 3}, \quad x > 0, x \in \mathbb{R} I(x)=x+340,x>0,x∈RA control system adjusts the distance x x\,x based on a setting s s\,s according to the function
g(s)=52lns,s>1,s∈R g(s) = \frac{5}{2} \ln s, \quad s > 1, s \in \mathbb{R} g(s)=25lns,s>1,s∈RDetermine, in simplest form, the value of the composite function Ig(e2)Ig(e^2)Ig(e2).
Find an expression for I−1(x)I^{-1}(x)I−1(x) and state its domain.
Hence, or otherwise, find all real solutions of the equation
I−1(x)=I(x) I^{-1}(x) = I(x) I−1(x)=I(x)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.