An environmental study models the concentration of a chemical byproduct, CCC, in mg/L, as a function of the distance xxx, in km, from the source of the spill. The concentration is given by the function:
C(x)=5x−2x−4,x>4 C(x) = \frac{5x - 2}{x - 4}, \quad x > 4 C(x)=x−45x−2,x>4Use calculus to prove that CCC is a decreasing function for its entire domain.
Find an expression for the inverse function C−1(x)C^{-1}(x)C−1(x).
Show that the composite function C(C(x))=ax+bx+cC(C(x)) = \frac{ax + b}{x + c}C(C(x))=x+cax+b where a,b,a, b,a,b, and ccc are constants to be determined.
Deduce the range of C(C(x))C(C(x))C(C(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.