8 The curve with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}
- The inverse of f is \(\mathrm { f } ^ { - 1 }\).
- Find \(\mathrm { f } ^ { - 1 } ( x )\).
- State the range of \(\mathrm { f } ^ { - 1 }\).
- Sketch, on the axes given on page 9 , the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
(2 marks)
- The function g is defined by
$$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
- Find \(\mathrm { gf } ( x )\), giving your answer in the form \(( a x - b ) ^ { 2 } - c\), where \(a , b\) and \(c\) are integers.
- Write down an expression for \(\mathrm { fg } ( x )\), and hence find the exact solution of the equation \(\operatorname { fg } ( x ) = \ln 5\).
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