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)
