Find range of composite

A question is this type if and only if it asks specifically for the range of a composite function like gf(x) or fg(x), requiring analysis of the composition.

4 questions · Standard +0.7

1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence
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Edexcel AEA 2012 June Q1
8 marks Challenging +1.2
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$
  1. Find the range of \(f\) . The function \(g\) is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$
  2. Find \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel AEA 2016 June Q1
7 marks Standard +0.8
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \quad x \in \mathbb { R } , x \geqslant 3$$
  1. Find the range of f . The function g is given by $$\operatorname { g } ( x ) = \frac { 10 } { x + 1 } \quad x \in \mathbb { R } , x \geqslant 4$$
  2. Find an expression for \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel Paper 2 2022 June Q10
8 marks Standard +0.3
  1. The function f is defined by
$$f ( x ) = \frac { 8 x + 5 } { 2 x + 3 } \quad x > - \frac { 3 } { 2 }$$
  1. Find \(\mathrm { f } ^ { - 1 } \left( \frac { 3 } { 2 } \right)\)
  2. Show that $$\mathrm { f } ( x ) = A + \frac { B } { 2 x + 3 }$$ where \(A\) and \(B\) are constants to be found. The function \(g\) is defined by $$g ( x ) = 16 - x ^ { 2 } \quad 0 \leqslant x \leqslant 4$$
  3. State the range of \(\mathrm { g } ^ { - 1 }\)
  4. Find the range of \(\mathrm { fg } ^ { - 1 }\)
SPS SPS FM 2025 October Q11
8 marks Standard +0.3
The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$
  1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
  2. Find the range of the function fg. [2]
  3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]