1. 1
   A printing company models poster costs using the function $f(x) = 3x - 5$.

   (a)

   A printing company models its cost per poster using the function $f(x) = 3x - 5$, where $x$ is the number of colours used. Find $f(4)$.

   [1 mark]

   • A7
   • B12
   • C9
   • D17

   (b)

   Using $f(x) = 3x - 5$, find the value of $x$ for which $f(x) = 16$.

   [1 mark]

   • A$x = 3.67$
   • B$x = 21$
   • C$x = 11$
   • D$x = 7$

   (c)

   Using $f(x) = 3x - 5$, evaluate $f(-2)$.

   [1 mark]

   • A$-1$
   • B$-11$
   • C$11$
   • D$1$

   Total for question 1: 3 marks

2. 2
   A scientist models particle energy using the function $h(x) = 2x^2 + 3$ and studies which input values are valid.

   (a)

   A scientist models the energy of a particle using the function $h(x) = 2x^2 + 3$, where $x$ is its speed in m/s. Find $h(3)$ and $h(-3)$, and state what you notice about the two results.

   [2 marks]

   (b)

   The scientist states that $h(x)$ is only used for speeds where $x \geq 0$. Describe, using a mapping diagram in words (i.e. describing which inputs map to which outputs, without drawing), how the inputs $x = 0, 1, 2$ map to their outputs under $h(x) = 2x^2+3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A currency exchange desk uses function notation to convert amounts and compute service charges.

   (a)

   A currency exchange desk uses the function $f(x) = \frac{x - 10}{4}$ to convert a fee-adjusted amount $x$ into a service charge. Find the inverse function $f^{-1}(x)$, showing full working.

   [3 marks]

   (b)

   Use your answer to part (a) to find $f^{-1}(6)$, and verify your answer by checking that $f(f^{-1}(6)) = 6$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A factory models production costs using two functions, $f(x)=2x+1$ and $g(x)=x^2$, combined and inverted for planning.

   (a)

   A factory uses two functions to model production costs: $f(x) = 2x + 1$ (raw material cost) and $g(x) = x^2$ (processing cost), where $x$ is the number of units. Find the composite function $gf(x) = g(f(x))$, showing full working, and evaluate $gf(3)$.

   [4 marks]

   (b)

   Find the composite function $fg(x) = f(g(x))$ and state whether $fg(x) = gf(x)$ in general, justifying your answer using the expressions you have found.

   [4 marks]

   (c)

   The factory manager needs to find the value of $x$ for which $fg(x) = 19$, using $fg(x) = 2x^2+1$ from part (b). Find the inverse function $f^{-1}(x)$ of $f(x)=2x+1$, then use it alongside solving the equation to find all valid values of $x$ (recall $x$ represents a number of units, so only non-negative solutions are valid).

   [5 marks]

   Total for question 4: 13 marks

End of questions