Piecewise function inverses

10 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662} The diagram shows the function f defined for \(- 1 \leqslant x \leqslant 4\), where $$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1 \\ \frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$

1. State the range of f .
2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
3. Obtain expressions to define the function \(\mathrm { f } ^ { - 1 }\), giving also the set of values for which each expression is valid.

7 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_778_816_255_662} The diagram shows the function f defined for \(0 \leqslant x \leqslant 6\) by $$\begin{aligned} & x \mapsto \frac { 1 } { 2 } x ^ { 2 } \quad \text { for } 0 \leqslant x \leqslant 2 , \\ & x \mapsto \frac { 1 } { 2 } x + 1 \text { for } 2 < x \leqslant 6 . \end{aligned}$$

1. State the range of f .
2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
3. Obtain expressions to define \(\mathrm { f } ^ { - 1 } ( x )\), giving the set of values of \(x\) for which each expression is valid.

7 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}

1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p \\ 5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.