1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} 5x - 2 & \text{for } x \leq 1 \\ -(x - 1)^2 + 1 & \text{for } 1 < x < 4 \\ \frac{3}{2}x - 13 & \text{for } x \geq 4 \end{cases}$$ We need to find the value of $f(4)$. 2. **Identify which piece to use:** Since $4 \geq 4$, we use the third piece: $$f(x) = \frac{3}{2}x - 13$$ 3. **Substitute $x=4$ into the third piece:** $$f(4) = \frac{3}{2} \times 4 - 13$$ 4. **Calculate the multiplication:** $$\frac{3}{2} \times 4 = \frac{3}{2} \times \cancel{4}^2 = 3 \times 2 = 6$$ 5. **Subtract 13:** $$f(4) = 6 - 13 = -7$$ **Final answer:** $$f(4) = -7$$