Can a function be its own inverse?

📌 CONCEPT: A function being its own inverse implies that the function is symmetric with respect to the line y = x, meaning the graph of the function and its inverse are identical.

📐 RULE / FORMULA: For a quadratic function f(x) = ax^2 + bx + c to be its own inverse, it must satisfy the condition f(f(x)) = x. This leads to the equation a^2 + 2a(b - 1) + (b^2 - 2b + c) = 0.

💡 WORKED EXAMPLE: Consider the quadratic function f(x) = x^2 - 2x + 1. To check if it's its own inverse, we substitute f(x) into the equation f(f(x)) = x and simplify. After solving, we find that f(x) = x^2 - 2x + 1 is its own inverse.

⚠️ COMMON MISTAKE: Students often assume that a function can be its own inverse only if it's a linear function, but in reality, a quadratic function can also be its own inverse if it satisfies the given conditions.