please explain how to solve this, its about differentiation

Question

given f(x) = -1/4 * ln3 * (x-1)^2 * (x-4)^2

and let us define g(x) = f(e^x)

the solution for the following eqution

g(g(x))=g(x) has 6 different solutions, why?

 

i guess the question is asking to draw the graph of g(x) and find the number of intersection for

y = g(g(x)) and y=g(x)

and i  have no idea how to approach it

Answer 1

You have a composition of functions.  g(x) composed of itself.

 

f(e^x) = g(x) = -(1/4)ln3(e^x - 1)²(e^x - 4)²

 

g( -(1/4)ln3(e^x - 1)²(e^x - 4)² ) = -(1/4)ln3(e^x - 1)²(e^x - 4)²

 

(-1/4)ln3(e^{-(1/4)ln3(e^x - 1)^2(e^x - 4)^2} - 1)²(e ^{-(1/4)ln3(e^x - 1)^2(e^x - 4)^2} - 4)² = -(1/4)ln3(e^x - 1)2(e^x - 4)²

 

This is end result equation.  Use it to find the number of zeros.