1. **State the problem:** A gardener has 18 metres of timber fencing to enclose a rectangular vegetable patch using a stone wall as one side. The width of the patch is $x$ metres. We need to show that the area $A(x)$ of the enclosure is given by the function $$A(x) = 18x - 2x^2.$$ 2. **Understand the setup:** The rectangle has one side formed by the stone wall, so fencing is needed for the other three sides: two widths and one length. 3. **Express the fencing constraint:** Let the length of the rectangle be $L$ metres. The total fencing used is for two widths and one length: $$2x + L = 18.$$ 4. **Solve for $L$:** $$L = 18 - 2x.$$ 5. **Write the area formula:** The area $A$ of the rectangle is width times length: $$A = x \times L = x(18 - 2x).$$ 6. **Expand the expression:** $$A = 18x - 2x^2.$$ 7. **Conclusion:** We have shown that the area of the enclosure is given by the function $$A(x) = 18x - 2x^2,$$ as required.