Functional Equation A8Ced4

1. **State the problem:** We are given the functional equation $$f(x) + f\left(1 - \frac{1}{x}\right) = 8x$$ and need to find the value of $$f(2)$$. 2. **Use the given equation:** Substitute $$x = 2$$: $$f(2) + f\left(1 - \frac{1}{2}\right) = 8 \times 2$$ which simplifies to $$f(2) + f\left(1 - 0.5\right) = 16$$ $$f(2) + f(0.5) = 16$$ 3. **Substitute $$x = 0.5$$:** $$f(0.5) + f\left(1 - \frac{1}{0.5}\right) = 8 \times 0.5$$ Calculate inside the function: $$1 - \frac{1}{0.5} = 1 - 2 = -1$$ So, $$f(0.5) + f(-1) = 4$$ 4. **Substitute $$x = -1$$:** $$f(-1) + f\left(1 - \frac{1}{-1}\right) = 8 \times (-1)$$ Calculate inside the function: $$1 - \frac{1}{-1} = 1 + 1 = 2$$ So, $$f(-1) + f(2) = -8$$ 5. **Now we have a system of three equations:** $$\begin{cases} f(2) + f(0.5) = 16 \\ f(0.5) + f(-1) = 4 \\ f(-1) + f(2) = -8 \end{cases}$$ 6. **Add all three equations:** $$[f(2) + f(0.5)] + [f(0.5) + f(-1)] + [f(-1) + f(2)] = 16 + 4 - 8$$ Simplify left side: $$2f(2) + 2f(0.5) + 2f(-1) = 12$$ Divide both sides by 2: $$f(2) + f(0.5) + f(-1) = 6$$ 7. **Use the first equation to substitute $$f(0.5) = 16 - f(2)$$ and the second to substitute $$f(-1) = 4 - f(0.5)$$:** Substitute $$f(0.5)$$ in $$f(-1)$$: $$f(-1) = 4 - (16 - f(2)) = 4 - 16 + f(2) = f(2) - 12$$ 8. **Substitute both into the sum:** $$f(2) + (16 - f(2)) + (f(2) - 12) = 6$$ Simplify: $$f(2) + 16 - f(2) + f(2) - 12 = 6$$ $$f(2) + 4 = 6$$ 9. **Solve for $$f(2)$$:** $$f(2) = 6 - 4 = 2$$ **Final answer:** $$\boxed{2}$$