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.
