1. **State the problem:** Solve the cubic equation $$f(x) = x^{3} - x - 1 = 0$$ for the roots. 2. **Formula and rules:** Cubic equations can have one or three real roots. We can use methods like the Rational Root Theorem to test simple roots or apply numerical methods for approximations. 3. **Check for rational roots:** Possible rational roots are factors of the constant term (±1) over factors of the leading coefficient (±1), so test $x=\pm1$. 4. **Evaluate at $x=1$:** $$1^{3} - 1 - 1 = 1 - 1 - 1 = -1 \neq 0$$ 5. **Evaluate at $x=-1$:** $$(-1)^{3} - (-1) - 1 = -1 + 1 - 1 = -1 \neq 0$$ No rational roots found. 6. **Use numerical approximation (e.g., Newton's method) or graph inspection:** The graph shows a root between 1 and 1.5. 7. **Approximate root:** Using numerical methods, the real root is approximately $$x \approx 1.3247$$. 8. **Summary:** The cubic equation has one real root near 1.3247 and two complex roots. **Final answer:** $$x \approx 1.3247$$