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)).