1. 1
   The function $f$ is defined by $f(x) = \dfrac{3x - 6}{x + 2}$, $x \in \mathbb{R}$, $x \ne -2$.

   (a)

   Write down the equation of the vertical asymptote of the graph of $f$.

   [1 mark]

   • A$x = 2$
   • B$x = -2$
   • C$x = 3$
   • D$y = -2$

   (b)

   Write down the equation of the horizontal asymptote of the graph of $f$.

   [1 mark]

   • A$y = -3$
   • B$y = \frac{1}{3}$
   • C$y = 0$
   • D$y = 3$

   (c)

   Find the coordinates of the points where the graph of $f$ meets the $x$-axis and the $y$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $h(x) = \dfrac{1}{x}$, $x \in \mathbb{R}$, $x \ne 0$.

   (a)

   Find the value of $(h \circ h)(5)$.

   [1 mark]

   • A$5$
   • B$\frac{1}{25}$
   • C$\frac{1}{5}$
   • D$25$

   (b)

   The point $\left(\frac{1}{4}, 4\right)$ lies on the graph of $y = h(x)$. Which other point must also lie on the graph, because $h$ is its own inverse?

   [1 mark]

   • A$\left(-\frac{1}{4}, 4\right)$
   • B$\left(\frac{1}{4}, -4\right)$
   • C$\left(4, \frac{1}{4}\right)$
   • D$(4, 4)$

   (c)

   Find the coordinates of the points where the graph of $y = h(x)$ meets the line $y = 4x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student practises touch-typing. After $t$ hours of practice, the student's typing speed, $W$ words per minute, is modelled by $W(t) = \dfrac{60t + 20}{t + 2}$, $t \ge 0$.

   (a)

   Find the number of hours of practice after which the model predicts a speed of 45 words per minute.

   [3 marks]

   (b)

   (i) Show that $W(t) = 60 - \dfrac{100}{t + 2}$.
   (ii) Hence write down the equation of the horizontal asymptote of the graph of $W$, and interpret it in the context of the model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x) = \dfrac{ax + 1}{2x - b}$, $x \ne \frac{b}{2}$, where $a, b \in \mathbb{R}$. The graph of $f$ has a vertical asymptote $x = 3$ and a horizontal asymptote $y = -2$.

   (a)

   (i) Find the value of $a$ and the value of $b$.
   (ii) Find the coordinates of the points where the graph of $f$ meets the axes.

   [6 marks]

   (b)

   (i) Find an expression for $f^{-1}(x)$ and state its domain.
   (ii) Write down the equations of the asymptotes of the graph of $y = f^{-1}(x)$, and explain how they are related to the asymptotes of the graph of $f$.

   [6 marks]

   Total for question 4: 12 marks

End of questions