3.13: Logarithmic inverse, sketch and nested square-root equation

0606/12/O/N/24 — Question 4 · 13 marks

The function ff is such that f(x)=4ln⁡(3x−2)f(x) = 4\ln(3x - 2), for x>ax > a, where aa is as small as possible.
(a)(i)
Write down the value of aa.
[1]
(a)(ii)
Write down the range of ff.
[1]
(a)(iii)
Find f−1(x)f^{-1}(x), stating its domain and range.
[4]
(a)(iv)
On the axes sketch the graphs of y=f(x)y = f(x) and y=f−1(x)y = f^{-1}(x), stating the intercepts with the axes.
[4]
(b)
Given that g(x)=(2x+1)12+4g(x) = (2x + 1)^{\frac{1}{2}} + 4, for x⩾0x \geqslant 0, solve the equation gg(x)=9gg(x) = 9.
[3]