1. **Problem 7: Write the piecewise function for** $$f(x) = \begin{cases} -x^2 + 5, & x < 2 \\ 5, & x = 2 \\ -3, & 2 < x < 5 \\ x - 2, & 5 \leq x \leq 8 \end{cases}$$ 2. **Problem 8: Write the piecewise function for** $$f(x) = \begin{cases} \sqrt{9 - (x+2)^2}, & -5 \leq x < 1 \\ -x - 1, & 1 \leq x < 4 \\ 2, & x \geq 4 \end{cases}$$ 3. **Problem 9: SUV value depreciation** - Initial value: 35750 - Decreases 2800 per year for first 4 years - Decreases 1700 per year for next 6 years Write piecewise function $V(x)$ for $0 \leq x \leq 10$: $$V(x) = \begin{cases} 35750 - 2800x, & 0 \leq x \leq 4 \\ 35750 - 2800 \times 4 - 1700(x - 4), & 4 < x \leq 10 \end{cases}$$ Simplify second piece: $$35750 - 11200 - 1700(x - 4) = 24550 - 1700x + 6800 = 31350 - 1700x$$ So, $$V(x) = \begin{cases} 35750 - 2800x, & 0 \leq x \leq 4 \\ 31350 - 1700x, & 4 < x \leq 10 \end{cases}$$ Calculate $V(3)$: $$V(3) = 35750 - 2800 \times 3 = 35750 - 8400 = 27350$$ Calculate $V(8)$: Since $8 > 4$, use second piece: $$V(8) = 31350 - 1700 \times 8 = 31350 - 13600 = 17750$$ 4. **Problem 10: Tax function** $$T(x) = \begin{cases} 0.15x, & 0 \leq x \leq 18000 \\ 0.28(x - 18000) + 2700, & 18000 < x \leq 44250 \\ 0.32(x - 44250) + 10050, & x > 44250 \end{cases}$$ Calculate $T(40000)$: Since $18000 < 40000 \leq 44250$, $$T(40000) = 0.28(40000 - 18000) + 2700 = 0.28 \times 22000 + 2700 = 6160 + 2700 = 8860$$ Interpretation: Tax on 40000 income is 8860. Calculate $T(75000)$: Since $75000 > 44250$, $$T(75000) = 0.32(75000 - 44250) + 10050 = 0.32 \times 30750 + 10050 = 9840 + 10050 = 19890$$ Interpretation: Tax on 75000 income is 19890.