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
    SECTION A: PURE MATHEMATICS

    The function f\mathrm{f} is defined by

    f(x)=2x+1x−3\mathrm{f}(x) = \dfrac{2x+1}{x-3}, x∈R\quad x \in \mathbb{R}, x≠3x \neq 3.
    (a) Find f−1(x)\mathrm{f}^{-1}(x), stating the value of xx for which f−1\mathrm{f}^{-1} is not defined. (3 marks)
    (b) Find ff(5)\mathrm{ff}(5). (2 marks)
    Level 2 · 5 marksTry it
  2. 2
    The functions f\text{f} and g\text{g} are defined by

    f(x)=x2−6x+14for x∈R\text{f}(x) = x^2 - 6x + 14 \quad \text{for } x \in \mathbb{R}

    g(x)=ln⁡(10−2x)\text{g}(x) = \ln(10 - 2x)

    where g\text{g} has its greatest possible domain.
    (a) Using set notation, state the range of f\text{f}

    (2 marks)
    (b) State the domain of g\text{g}

    (1 mark)
    Level 2 · 3 marksTry it
  3. 3
    The function ff is defined by

    f(x)=3x+1x−4x∈R, x>4f(x) = \frac{3x + 1}{x - 4} \qquad x \in \mathbb{R},\ x > 4
    (a) State the range of ff.

    (1 mark)
    (b) Find f−1(x)f^{-1}(x), stating the domain of f−1f^{-1}.

    (3 marks)
    Level 2 · 4 marksTry it