19. The function f is defined by \(\mathrm { f } : x \mapsto \frac { 3 x - 1 } { x - 3 } , x \in \mathbb { R } , x \neq 3\).
- Prove that \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) for all \(x \in \mathbb { R } , x \neq 3\).
- Hence find, in terms of \(k , \operatorname { ff } ( k )\), where \(x \neq 3\).
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Figure 3 shows a sketch of the one-one function g , defined over the domain \(- 2 \leq x \leq 2\). - Find the value of \(\mathrm { fg } ( - 2 )\).
- Sketch the graph of the inverse function \(\mathrm { g } ^ { - 1 }\) and state its domain.
The function h is defined by \(\mathrm { h } : x \mapsto 2 \mathrm {~g} ( x - 1 )\).
- Sketch the graph of the function h and state its range.