1. 1
   Let $f(x) = 2^{x}$, $x \in \mathbb{R}$, and $g(x) = \log_{2} x$, $x > 0$.

   (a)

   Find the value of $g\left(\frac{1}{8}\right)$.

   [1 mark]

   • A$3$
   • B$-3$
   • C$-\frac{1}{3}$
   • D$\frac{1}{16}$

   (b)

   Which of the following is equal to $f(x)$ for all real $x$?

   [1 mark]

   • A$e^{x \ln 2}$
   • B$2e^{x}$
   • C$e^{2x}$
   • D$e^{2 \ln x}$

   (c)

   Write down the range of $f$. Hence explain how it is related to the domain of $g$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $q(x) = \log_{3} x$, $x > 0$.

   (a)

   Find the value of $q\left(\sqrt{3}\right)$.

   [1 mark]

   • A$2$
   • B$-\frac{1}{2}$
   • C$\frac{1}{2}$
   • D$\frac{\sqrt{3}}{3}$

   (b)

   Solve the equation $q(x) = -2$.

   [1 mark]

   • A$x = -9$
   • B$x = 9$
   • C$x = -6$
   • D$x = \frac{1}{9}$

   (c)

   Find the exact value of $q(81) + 3^{q(5)}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A laboratory stores a sample of a radioactive isotope. The mass, $m$ grams, remaining $t$ days after the sample is first measured is modelled by $m(t) = 80 \times 2^{-\frac{t}{12}}$, $t \ge 0$.

   (a)

   Find the mass remaining after 36 days, and find the time taken for the mass to fall to 5 grams.

   [3 marks]

   (b)

   (i) Show that $m(t) = 80e^{-kt}$, where $k = \dfrac{\ln 2}{12}$.
   (ii) Write down the equation of the horizontal asymptote of the graph of $m$, and interpret it in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x) = 2e^{x} + 1$, $x \in \mathbb{R}$.

   (a)

   (i) Write down the equation of the horizontal asymptote of the graph of $f$, and the range of $f$.
   (ii) Find $f^{-1}(x)$ and state its domain.

   [6 marks]

   (b)

   (i) Write down the equation of the vertical asymptote of the graph of $y = f^{-1}(x)$, and find the coordinates of the point where this graph meets the $x$-axis.
   (ii) Given that $f(\ln k) = 11$, find the value of $k$. Hence write down the exact value of $f^{-1}(11)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions