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)
