The signal strength fff, measured in decibels (dB), of a satellite transmitter is modeled as a function of the distance x x\,x from the source (in km) by
f(x)=56x+1,x>0,x∈R f(x) = \frac{56}{x + 1}, \quad x > 0, x \in \mathbb{R} f(x)=x+156,x>0,x∈RThe distance x x\,x is determined by the signal frequency ω\omegaω (in MHz) according to the function
g(ω)=23lnω,ω>1,ω∈R g(\omega) = \frac{2}{3} \ln \omega, \quad \omega > 1, \omega \in \mathbb{R} g(ω)=32lnω,ω>1,ω∈RFind, in simplest form, the value of fg(e9)fg(e^9)fg(e9).
Find an expression for f−1(x)f^{-1}(x)f−1(x) and state its domain.
Hence, or otherwise, find all real solutions of the equation
f−1(x)=f(x) f^{-1}(x) = f(x) f−1(x)=f(x)532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.