Functions: past paper questions

Where the questions come from

• Question 4Oct/Nov 2025, paper 11Structured6 marks

  Express $1-6x-x^2$ in the form $a-(x+b{)}^2$, where $a$ and $b$ are constants.

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• Question 5Oct/Nov 2024, paper 12Structured10 marksHas a diagram

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{2x+1}{2x-1}$ for $x<\frac{1}{2}$. State the value of $\mathrm{f}(-1)$.

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• Question 9Oct/Nov 2023, paper 11Structured9 marks

  Express $4x^2-12x+13$ in the form $(2x+a{)}^2+b$, where $a$ and $b$ are constants.

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• Question 2May/June 2026, paper 12Structured7 marksHas a diagram

  In the diagram, the graph with equation $y=\mathrm{f}(x)$ is shown with solid lines and the graph with equation $y=\mathrm{g}(x)$ is shown with broken lines.

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• Question 3Oct/Nov 2025, paper 12Structured5 marks

  The graph of $y=\mathrm{f}(x)$ is transformed to the graph of $y=\mathrm{f}(3x)+2$. Describe fully the two transformations which have been combined to give the resulting graph.

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• Question 5Oct/Nov 2024, paper 13Structured8 marksHas a diagram

  In the diagram, the graph with equation $y=\mathrm{f}(x)$ is shown with solid lines and the graph with equation $y=\mathrm{g}(x)$ is shown with broken lines.

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• Question 6Oct/Nov 2023, paper 12Structured8 marks

  The equation of a curve is $y=x^2-8x+5$. Find the coordinates of the minimum point of the curve.

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• Question 9Oct/Nov 2025, paper 12Structured8 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{4}{(3x-6{)}^2}+\frac{1}{(3x-6{)}^3}$ for $x>2$. Find an expression for ${\mathrm{f}}^{\mathrm{\prime }}(x)$ and hence determine whether $\mathrm{f}$ is an increasing function, a decreasing function or neither.

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• Question 8Oct/Nov 2024, paper 13Structured9 marksHas a diagram

  Express $3x^2-12x+14$ in the form $3(x+a{)}^2+b$, where $a$ and $b$ are constants to be found.

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• Question 6May/June 2026, paper 13Structured7 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{16}{x^3}+3x+5$ for $x\mathrm{\ne }0$. Find the coordinates of all stationary points on the curve $y=\mathrm{f}(x)$.

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• Question 6Oct/Nov 2025, paper 13Structured8 marks

  The diagram shows the graphs of $y=x^3$ and $y=\mathrm{f}(x)$. The graph of $y=x^3$ is transformed to the graph of $y=\mathrm{f}(x)$ by a sequence of transformations. Describe fully a suitable sequence of transformations.

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• Question 2May/June 2024, paper 11Structured6 marksHas a diagram

  The diagram shows two curves. One curve has equation $y=\sin x$ and the other curve has equation $y=\mathrm{f}(x)$. In order to transform the curve $y=\sin x$ to the curve $y=\mathrm{f}(x)$, the curve $y=\sin x$ is first reflected in the $x$-axis.

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• Question 7Oct/Nov 2023, paper 13Structured7 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=1+\frac{3}{x-2}$ for $x>2$. State the range of $\mathrm{f}$.

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• Question 7Oct/Nov 2025, paper 13Structured4 marks

  The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=\frac{2}{ax-3}+\frac{1}{2}$ for $x>\frac{3}{a}$, where $a$ is a positive constant. Find ${\mathrm{g}}^{-1}(x)$ and hence verify that if $a=6$ then ${\mathrm{g}}^{-1}(x)\equiv \mathrm{g}(x)$.

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• Question 6May/June 2024, paper 11Structured7 marksHas a diagram

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{2}{x^2}+4$ for $x<0$. The diagram shows the graph of $y=\mathrm{f}(x)$. On this diagram, sketch the graph of $y={\mathrm{f}}^{-1}(x)$.

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• Question 8Oct/Nov 2023, paper 13Structured5 marksHas a diagram

  The diagram shows part of the graph of $y=\sin (a(x+b))$, where $a$ and $b$ are positive constants. State the value of $a$ and one possible value of $b$.

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• Question 2May/June 2024, paper 12Structured5 marks

  The curve $y=x^2$ is transformed to the curve $y=4(x-3{)}^2-8$. Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied.

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• Question 3May/June 2023, paper 11Structured4 marksHas a diagram

  The diagram shows graphs with equations $y=\mathrm{f}(x)$ and $y=\mathrm{g}(x)$. Describe fully a sequence of two transformations which transforms the graph of $y=\mathrm{f}(x)$ to $y=\mathrm{g}(x)$.

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• Question 4May/June 2024, paper 12Structured7 marksHas a diagram

  The function $\mathrm{f}$ is defined as follows: $$\mathrm{f}(x)=\sqrt{x-1}\text{ for }x>1.$$ Find an expression for ${\mathrm{f}}^{-1}(x)$.

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• Question 8May/June 2023, paper 11Structured8 marks

  The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows, where $a$ and $b$ are constants.

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• Question 1May/June 2025, paper 12Structured4 marksHas a diagram

  The diagram shows the graphs with equations $y=\mathrm{f}(x)$ and $y=\mathrm{g}(x)$. Describe fully a sequence of two transformations which transforms the graph of $y=\mathrm{f}(x)$ to the graph of $y=\mathrm{g}(x)$.

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• Question 6May/June 2024, paper 13Structured7 marks

  A curve passes through the point $(\frac{4}{5},-3)$ and is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-20}{(5x-3{)}^2}$. Find the equation of the curve.

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• Question 11May/June 2025, paper 12Structured9 marks

  Express $x^2+4x+2$ in the form $(x+a{)}^2+b$, where $a$ and $b$ are integers.

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• Question 1May/June 2023, paper 13Structured4 marksHas a diagram

  The diagram shows the graph of $y=\mathrm{f}(x)$, which consists of the two straight lines $AB$ and $BC$.

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• Question 10May/June 2025, paper 13Structured11 marks

  A curve $C$ has equation $y=\frac{9}{2x-5}+2x-5$. Find the coordinates of the two stationary points.

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• Question 7May/June 2023, paper 13Structured8 marks

  The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2-\frac{5}{x+2}$ for $x>-2$. State the range of $\mathrm{f}$.

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