Can a Function be its own Inverse?

📌 CONCEPT: A function can be its own inverse if it is a bijective function, meaning it is both one-to-one and onto, and f(f(x)) = x for all x in the domain of f.

📐 RULE / FORMULA: The function must satisfy the condition f(f(x)) = x, which implies that f(x) must be a bijection.

💡 WORKED EXAMPLE: Consider the function f(x) = x^3, which is a bijection. To verify if it is its own inverse, we need to check if f(f(x)) = x. We have f(f(x)) = f(x^3) = (x^3)^3 = x^9, which is not equal to x. However, if we consider the function f(x) = x^1/3, which is also a bijection, we have f(f(x)) = f(x^1/3) = (x^1/3)^1/3 = x.

⚠️ COMMON MISTAKE: Students may assume that any function can be its own inverse, but in reality, it requires the function to be a bijection, which is a specific property.