Mathematics

Problem

Define the functions

f(x)=\dfrac{1}{\sqrt{3x-2}}

with domain

(\tfrac{2}{3},\infty)

and

g(x)=\ln x

with domain

(0,\infty)

. a. Part 1: (i) Write the explicit form of the composite

(g\circ f)(x)

, simplifying as far as possible. b. Part 2: (ii) Determine the domain of

(g\circ f)

, showing each step of your working. c. Part 3: (iii) Solve the inequality

(g\circ f)(x)>0

and give the solution set in interval form. d. Part 4: (iv) Prove that

(g\circ f)(x)

is a decreasing function on its domain.