Inverse of a Function Exists?

📌 CONCEPT: A function has an inverse if and only if it is one-to-one, meaning no two different elements in the domain have the same element in the range.

📐 RULE / FORMULA: For a function to have an inverse, it must pass the horizontal line test, where no horizontal line intersects the graph of the function in more than one place.

💡 WORKED EXAMPLE: Consider the function f(x) = x^2. It is observed that no two different elements of the domain have the same element in the range. However, if we consider the function g(x) = x^2, where g(x) = f(x) for x < 0 and g(x) = -f(x) for x ≥ 0, then g(x) is not one-to-one. Hence, the inverse of f(x) does not exist.

⚠️ COMMON MISTAKE: Students often mistakenly assume that a function has an inverse if it is observed that no two different elements of the domain have the same element in the range. However, this observation is necessary but not sufficient for a function to have an inverse.