Delta Cycle Works prices every repair with the number machine shown below. The input is the number of hours x that the job takes on the bench. The output is...

A number machine is a picture of a function. The boxes are applied left to right, so the input x is first multiplied by 18 and only then is 25 added. Reading the machine in that order is the whole skill being tested here: the multiplication is the hourly labour rate and the addition is a fixed charge that does not depend on how long the job takes.

1. Price of a 3 hour job [1 mark]
   Feed \(x = 3\) through the two boxes in turn: \[3 \times 18 = 54, \qquad 54 + 25 = 79\] The price is £79. Notice that the answer is not \(3 \times 43\); the 25 is added once, at the end, not to every hour.
2. f(x) in terms of x [1 mark]
   Writing the same two steps algebraically gives \[f(x) = 18x + 25\] This is a linear function: 18 is the gradient (the cost of each extra hour on the bench) and 25 is the intercept (the standing charge).
3. Hours for a £142 repair [2 marks]
   Here the output is known and the input is wanted, so the equation is solved rather than evaluated: \[18x + 25 = 142 \quad \Rightarrow \quad 18x = 117 \tag{1 mark}\] \[x = \frac{117}{18} = 6.5 \tag{1 mark}\] The job took 6.5 hours. Undoing the machine in reverse order gives the same result: subtract 25 first, then divide by 18.
4. The inverse function [1 mark]
   The inverse reverses the machine, so the boxes are run backwards and each operation is replaced by its opposite: subtract 25, then divide by 18. \[f^{-1}(x) = \frac{x - 25}{18}\] Algebraically, set \(y = 18x + 25\), rearrange to \(x = \dfrac{y - 25}{18}\) and then swap the letters. A common slip is to write \(\dfrac{x}{18} - 25\), which divides before subtracting and so undoes the machine in the wrong order. Check it works: \(f^{-1}(142) = \dfrac{117}{18} = 6.5\), matching part (c).

Examination reminder: with a number machine, always list the operations in order first, then reverse both the order and the operations to build the inverse.