Challenging +1.2 This is a multi-part question on rational function inverses and compositions that requires systematic algebraic manipulation and careful domain tracking. Part (a) is routine inverse finding with parameters. Parts (b-d) involve iterating a specific rational function and finding patterns, which requires more work but follows standard techniques. Part (e) combines g and its inverse in an inequality, requiring algebraic skill but no deep insight. The length and iteration aspects push it above average difficulty, but the techniques are all standard A-level pure maths.
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 \(h ( x ) \geq 4\). [0pt]
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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 $h ( x ) \geq 4$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q6 [20]}}