6. A function is defined by: $$f ( x ) = \frac { a x + b } { c x + d } , x \in \mathbb { R } , x \neq - \frac { d } { c }$$ a) Find and simplify an expression for \(f ^ { - 1 } ( x )\), stating the domain. A function is defined by: $$g ( x ) = \frac { x - 6 } { x - 4 } , x \in \mathbb { R } , x \neq 4$$ b) Find \(g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\), stating an appropriate domain for each function.
c) Find \(g ^ { - 1 } ( x ) , g ^ { - 2 } ( x )\) and \(g ^ { - 3 } ( x )\), stating an appropriate domain for each function. NB: \(g ^ { - n } ( x ) = g ^ { - 1 } \left( g ^ { - 1 } \left( \cdots \left( g ^ { - 1 } ( x ) \right) \cdots \right) \right)\) with \(n\) copies of \(g ^ { - 1 }\).
d) State the range of \(g ( x ) , g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\). A function is defined (over an appropriate domain) by \(h ( x ) = g ( x ) + g ^ { - 1 } ( x )\).
e) Solve the inequality \(h ( x ) \geq 4\).
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