1. **State the problem:** We need to write a piecewise function $T(x)$ that gives the tax owed based on taxable income $x$ for $x < 117950$. 2. **Given tax brackets:** - For $0 < x \leq 23350$, tax is 15% of $x$. - For $23350 < x \leq 56550$, tax is $3502.50 + 28\%$ of the amount over $23350$. - For $56550 < x < 117950$, tax is $12798.50 + 31\%$ of the amount over $56550$. 3. **Write the piecewise function:** $$ T(x) = \begin{cases} 0.15x & \text{if } 0 < x \leq 23350 \\ 3502.50 + 0.28(x - 23350) & \text{if } 23350 < x \leq 56550 \\ 12798.50 + 0.31(x - 56550) & \text{if } 56550 < x < 117950 \end{cases} $$ 4. **Evaluate $T(31950)$:** Since $23350 < 31950 \leq 56550$, use the second case: $$ T(31950) = 3502.50 + 0.28(31950 - 23350) $$ Calculate the amount over $23350$: $$ 31950 - 23350 = 8600 $$ Calculate the tax on the amount over: $$ 0.28 \times 8600 = 2408 $$ Add the base tax: $$ T(31950) = 3502.50 + 2408 = 5910.50 $$ **Final answers:** - A. The piecewise function $T(x)$ is as above. - B. The tax owed for $x=31950$ is $5910.50$.