Define the tax function T(x) for each income bracket before the rebate.

Step 1: Define the tax function $T(x)$ for each income bracket before the rebate.
Let $x$ be the annual income.

• For income less than R55 000: No tax is paid. $T_{gross}(x) = 0$ for $x < 55000$.

• For income between R55 000 and R85 000 (inclusive): Tax is 9% of the total income. $T_{gross}(x) = 0.09x$ for $55000 \le x \le 85000$.

• For income above R85 000: Tax consists of a fixed