A-level · Algebra & functions

Composite and inverse functionsExam-style questions

Chaining functions together and running them backwards. Composite functions, read inside-out, and why the order matters; solving an equation with a composite, and the domain check it needs; finding an inverse by rearranging, and why it is not a reciprocal; the inverse's graph as a reflection in y = x; the domain and range swapping over; and why only a one-to-one function has an inverse.

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42 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    The function f\mathrm{f} is defined by

    f(x)=3−2e−x,x∈R\mathrm{f}(x) = 3 - 2\mathrm{e}^{-x}, \qquad x \in \mathbb{R}
    (a) State the range of f\mathrm{f}. (2 marks)
    (b) Find f−1(x)\mathrm{f}^{-1}(x) and state its domain. (4 marks)
    Level 4 · 6 marksTry it
  2. 2
    The function ff is defined by f(x)=2x+3x−1f(x) = \dfrac{2x + 3}{x - 1},   x≠1\; x \neq 1.

    Find f−1(x)f^{-1}(x), stating the value of xx for which f−1f^{-1} is not defined. (4 marks)
    Level 4 · 4 marksTry it
  3. 3
    The function ff is defined by f(x)=x2−4x+7f(x) = x^{2} - 4x + 7 for x⩾1x \geqslant 1.
    (a) Write f(x)f(x) in the form (x−a)2+b(x - a)^{2} + b. (2 marks)
    (b) Hence find the range of ff. (2 marks)
    Level 4 · 4 marksTry it
  4. 4
    The functions f\mathrm{f} and g\mathrm{g} are defined by

    f(x)=3x−1x+2,x∈R,  x≠−2\mathrm{f}(x) = \dfrac{3x - 1}{x + 2}, \qquad x \in \mathbb{R},\; x \neq -2

    g(x)=2x+5,x∈R\mathrm{g}(x) = 2x + 5, \qquad x \in \mathbb{R}
    (a) Find fg(x)\mathrm{fg}(x), giving your answer as a single fraction in its simplest form, and state the value of xx for which fg(x)\mathrm{fg}(x) is undefined. (3 marks)
    (b) Find f−1(x)\mathrm{f}^{-1}(x), and state its domain. (4 marks)
    (c) Explain why the equation f(x)=3\mathrm{f}(x) = 3 has no solutions. (1 mark)
    Level 4 · 8 marksTry it
  5. 5
    The function ff is defined by f(x)=x+4x−1f(x) = \dfrac{x + 4}{x - 1}, x≠1x \ne 1. Find f−1(x)f^{-1}(x).
    Level 4 · 3 marksTry it

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