Find inverse function after simplification

Simplify a complex rational function to a simpler form, then find its inverse function and possibly its domain/range.

7 questions

Edexcel P3 2021 October Q1
  1. The function f is defined by
$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }\)
  2. Find \(\mathrm { f } ^ { - 1 }\)
    1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.
      (3)
Edexcel C3 2012 January Q7
  1. The function f is defined by
$$\mathrm { f } : x \mapsto \frac { 3 ( x + 1 ) } { 2 x ^ { 2 } + 7 x - 4 } - \frac { 1 } { x + 4 } , \quad x \in \mathbb { R } , x > \frac { 1 } { 2 }$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 x - 1 }\)
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\)
  3. Find the domain of \(\mathrm { f } ^ { - 1 }\) $$\mathrm { g } ( x ) = \ln ( x + 1 )$$
  4. Find the solution of \(\mathrm { fg } ( x ) = \frac { 1 } { 7 }\), giving your answer in terms of e .
Edexcel C3 2005 June Q3
3. The function \(f\) is defined by $$f : x \rightarrow \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } : x \rightarrow x ^ { 2 } + 5 , \quad x \in \mathbb { R }$$
  3. Solve \(\operatorname { fg } ( x ) = \frac { 1 } { 4 }\).
Edexcel C3 2014 June Q5
5. $$\mathrm { g } ( x ) = \frac { x } { x + 3 } + \frac { 3 ( 2 x + 1 ) } { x ^ { 2 } + x - 6 } , \quad x > 3$$
  1. Show that \(\mathrm { g } ( x ) = \frac { x + 1 } { x - 2 } , \quad x > 3\)
  2. Find the range of g.
  3. Find the exact value of \(a\) for which \(\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )\).
Edexcel C3 2018 June Q2
  1. The function f is defined by
$$\mathrm { f } ( x ) = \frac { 6 } { 2 x + 5 } + \frac { 2 } { 2 x - 5 } + \frac { 60 } { 4 x ^ { 2 } - 25 } , \quad x > 4$$
  1. Show that \(\mathrm { f } ( x ) = \frac { A } { B x + C }\) where \(A , B\) and \(C\) are constants to be found.
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
Edexcel C3 Q7
7. \(\quad \mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1\)
  1. Prove that \(\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }\).
  2. Find the range of f.
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  4. Find the range of \(\mathrm { f } ^ { - 1 } ( x )\).
Edexcel C3 Q7
7. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 } , \quad x \in \mathbb { R } , \quad x < 1$$
  1. Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
  2. Find an expression for the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 2\).