5 The function f is defined by
$$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { 3 } , \text { for real values of } x , \text { where } \boldsymbol { x } \leqslant \mathbf { 0 }$$
- State the range of f.
- The inverse of f is \(\mathrm { f } ^ { - 1 }\).
- Write down the domain of \(\mathrm { f } ^ { - 1 }\).
- Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
- The function g is defined by
$$\mathrm { g } ( x ) = \ln | 3 x - 1 | , \quad \text { for real values of } x , \text { where } x \neq \frac { 1 } { 3 }$$
The curve with equation \(y = \mathrm { g } ( x )\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-6_469_819_1254_612}
- The curve \(y = \mathrm { g } ( x )\) intersects the \(x\)-axis at the origin and at the point \(P\).
Find the \(x\)-coordinate of \(P\).
- State whether the function \(g\) has an inverse. Give a reason for your answer.
- Show that \(\operatorname { gf } ( x ) = \ln \left| x ^ { 2 } - k \right|\), stating the value of the constant \(k\).
- Solve the equation \(\mathrm { gf } ( x ) = 0\).