1. 1
   The function $f$ is defined by $f(x) = 2\sqrt{x-3} + 1$, for $x \ge 3$.

   (a)

   Find the range of $f$.

   [1 mark]

   • A$f(x) \ge 3$
   • B$f(x) \ge 0$
   • C$f(x) \ge 1$
   • D$f(x) \in \mathbb{R}$

   (b)

   Find $f^{-1}(7)$.

   [1 mark]

   • A$5$
   • B$12$
   • C$3$
   • D$\frac{1}{5}$

   (c)

   Write down the domain and the range of $f^{-1}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A taxi company charges a fixed fee of 3.50 dollars plus 1.80 dollars per kilometre. The cost, in dollars, of a journey of $d$ kilometres is modelled by $C(d) = 3.5 + 1.8d$, where $0 < d \le 40$.

   (a)

   Find $C(12)$.

   [1 mark]

   • A$43.80$
   • B$21.60$
   • C$4.72$
   • D$25.10$

   (b)

   A passenger pays 48.50 dollars for a journey. Find the length of the journey in kilometres.

   [1 mark]

   • A$25$
   • B$26.9$
   • C$28.9$
   • D$90.8$

   (c)

   Find the range of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $g$ is defined by $g(x) = (x-2)^2 + 1$, for $x \in \mathbb{R}$.

   (a)

   Show that $g$ is not a one-to-one function, and hence explain why $g$ does not have an inverse.

   [3 marks]

   (b)

   The domain of $g$ is now restricted to $x \ge k$, where $k$ is as small as possible, so that the inverse function $g^{-1}$ exists. Write down the value of $k$ and the domain of $g^{-1}$. Hence find $g^{-1}(10)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $F(c) = 1.8c + 32$ converts a temperature of $c$ degrees Celsius into degrees Fahrenheit. A particular oven operates at temperatures from $20\,^{\circ}\mathrm{C}$ to $250\,^{\circ}\mathrm{C}$ inclusive, so for this oven the domain of $F$ is $20 \le c \le 250$.

   (a)

   (i) Find the range of $F$ for this oven.
   (ii) Find $F^{-1}(x)$ and state its domain.
   (iii) A recipe says to bake at $356\,^{\circ}\mathrm{F}$. Use your answer to (ii) to find the setting in degrees Celsius.

   [6 marks]

   (b)

   Ignore the oven's restriction for this part, so that $F$ and $F^{-1}$ are defined for all real numbers.
   (i) Show that the graphs of $y = F(x)$ and $y = F^{-1}(x)$ intersect at the point $(-40, -40)$.
   (ii) Now use the oven domain again. Hence explain why, for the oven, the graphs of $y = F(x)$ and $y = F^{-1}(x)$ do not intersect.

   [6 marks]

   Total for question 4: 12 marks

End of questions