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

   (a)

   Find $(f \circ g)(2)$.

   [1 mark]

   • A$27$
   • B$30$
   • C$17$
   • D$11$

   (b)

   Find $(g \circ f)(x)$.

   [1 mark]

   • A$3x^2 + 5$
   • B$9x^2 - 6x + 3$
   • C$9x^2 + 3$
   • D$3x^3 - x^2 + 6x - 2$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $h$ is defined by $h(x) = \frac{2x + 3}{x - 2}$, for $x \in \mathbb{R}$, $x \ne 2$.

   (a)

   Find $(h \circ h)(4)$.

   [1 mark]

   • A$30.25$
   • B$5.5$
   • C$\frac{2}{11}$
   • D$4$

   (b)

   It is given that $(h \circ h)(x) = x$ for all $x \ne 2$. Find $h^{-1}(5)$.

   [1 mark]

   • A$\frac{13}{3}$
   • B$\frac{3}{13}$
   • C$5$
   • D$\frac{5}{3}$

   (c)

   Using the fact that $h^{-1} = h$, write down the range of $h$. Give a reason for your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The functions $f$ and $g$ are defined by $f(x) = \sqrt{x + 5}$, for $x \ge -5$, and $g(x) = x^2 - 9$, for $x \in \mathbb{R}$.

   (a)

   Find $(f \circ g)(x)$, and state its domain.

   [3 marks]

   (b)

   Show that $(g \circ f)(x) = x - 4$, and state the domain of $g \circ f$. Hence explain why $g \circ f$ is not the same function as $k(x) = x - 4$, $x \in \mathbb{R}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Bank A changes UAE dirhams (AED) into euros. It first deducts a fee of 15 AED, modelled by $f(x) = x - 15$, and then converts the remaining dirhams at 0.24 euros per dirham, modelled by $g(x) = 0.24x$. Bank B converts all of the dirhams at 0.245 euros per dirham and then deducts a fee of 5 euros. In both banks $x$ is the number of dirhams handed over, where $x > 50$.

   (a)

   (i) Find an expression for $(g \circ f)(x)$.
   (ii) Find the number of euros that Bank A gives for 1000 AED.
   (iii) Find $(g \circ f)^{-1}(x)$. Hence find the number of dirhams needed at Bank A to receive 300 euros.

   [6 marks]

   (b)

   (i) Write Bank B's conversion as a composite function $(q \circ p)(x)$, stating $p(x)$ and $q(x)$, and simplify it.
   (ii) Find the amounts of dirhams for which Bank B gives more euros than Bank A.
   (iii) Explain, in context, why $(f \circ g)(x)$ does not model Bank A.

   [6 marks]

   Total for question 4: 12 marks

End of questions