SUMMARY
The inverse of the function f(x) = x² - 8x + 8, defined for x ≤ 4, is determined by first setting y = x² - 8x + 8 and then switching x and y. This leads to the equation x = y² - 8y + 8. By rearranging and completing the square, the equation simplifies to (y - 4)² = x + 8. The final expression for the inverse function is f⁻¹(x) = 4 ± √(x + 8), with the domain of f⁻¹(x) being x ≥ -8, reflecting the range of the original function f(x).
PREREQUISITES
- Understanding of quadratic functions and their properties
- Familiarity with the concept of inverse functions
- Knowledge of completing the square technique
- Proficiency in using the quadratic formula
NEXT STEPS
- Study the properties of inverse functions in detail
- Learn how to complete the square for various quadratic equations
- Explore the quadratic formula and its applications in solving equations
- Investigate the relationship between the domain and range of functions and their inverses
USEFUL FOR
Students studying algebra, mathematicians interested in function analysis, and educators teaching inverse functions and quadratic equations.