1. **Problem 3:** Find the inverse of the function $g(x) = -3 + (x - 1)^3$ and graph both the function and its inverse. 2. **Step 1:** Write the function as $y = -3 + (x - 1)^3$. 3. **Step 2:** To find the inverse, swap $x$ and $y$: $$x = -3 + (y - 1)^3$$ 4. **Step 3:** Solve for $y$: $$x + 3 = (y - 1)^3$$ $$y - 1 = \sqrt[3]{x + 3}$$ $$y = 1 + \sqrt[3]{x + 3}$$ 5. **Step 4:** The inverse function is: $$g^{-1}(x) = 1 + \sqrt[3]{x + 3}$$ --- 6. **Problem 4:** Solve the equation $6 \cdot 5^{a + 3} = 77$ for $a$, rounding to the nearest ten-thousandth. 7. **Step 1:** Isolate the exponential term: $$5^{a + 3} = \frac{77}{6}$$ 8. **Step 2:** Take the logarithm base 5 of both sides: $$a + 3 = \log_5 \left( \frac{77}{6} \right)$$ 9. **Step 3:** Use change of base formula: $$a + 3 = \frac{\ln \left( \frac{77}{6} \right)}{\ln 5}$$ 10. **Step 4:** Calculate the right side: $$\frac{\ln \left( \frac{77}{6} \right)}{\ln 5} = \frac{\ln(12.8333)}{\ln 5} \approx \frac{2.5502}{1.6094} \approx 1.5843$$ 11. **Step 5:** Solve for $a$: $$a = 1.5843 - 3 = -1.4157$$ --- **Final answers:** - Inverse function: $$g^{-1}(x) = 1 + \sqrt[3]{x + 3}$$ - Solution for $a$: $$a \approx -1.4157$$