1. The problem is to express the tax payable $T$ as a function of taxable income $I$ based on the given income ranges and tax rules. 2. Define $T(I)$ in pieces according to the income brackets: - For $0 \leq I \leq 6000$: $$T(I) = 0$$ - For $6001 \leq I \leq 20000$: $$T(I) = 0.17 \times (I - 6000)$$ - For $20001 \leq I \leq 50000$: $$T(I) = 2380 + 0.30 \times (I - 20000)$$ - For $50001 \leq I \leq 75000$: $$T(I) = 11380 + 0.42 \times (I - 50000)$$ - For $I > 75000$: $$T(I) = 21880 + 0.48 \times (I - 75000)$$ 3. This piecewise function allows calculating tax for any income based on the corresponding bracket. 4. This function is continuous at the boundary points (6000, 20000, 50000, 75000) confirming correctness. Final answer: $$ T(I) = \begin{cases} 0 & 0 \leq I \leq 6000 \\ 0.17 (I - 6000) & 6001 \leq I \leq 20000 \\ 2380 + 0.30 (I - 20000) & 20001 \leq I \leq 50000 \\ 11380 + 0.42 (I - 50000) & 50001 \leq I \leq 75000 \\ 21880 + 0.48 (I - 75000) & I > 75000 \end{cases} $$