- The function \(f\) is defined by
$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$
- Show that the inverse function of f is defined by
$$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$
and write down the domain of \(\mathrm { f } ^ { - 1 }\).
- Write down the range of \(\mathrm { f } ^ { - 1 }\).
- In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes.
The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\).
The iterative formula
$$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$
is used to find an approximate value for \(k\).
- Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
- Find the value of \(k\) to 3 decimal places.