Relational Mapping Conundrum?

📌 CONCEPT: When two functions are composed with each other, the resulting relation between their domains and codomains can be either a function or a relation, depending on the properties of the individual functions.

📐 RULE / FORMULA: The composite function f ∘ g is a function if the range of g is a subset of the domain of f.

💡 WORKED EXAMPLE: Let's consider f(x) = 2x and g(x) = x^2. When we compose f with g, we get f ∘ g(x) = 2(x^2) = 2x^2. Since the range of g is all real numbers and the domain of f is all real numbers, the composite function f ∘ g is a function.

⚠️ COMMON MISTAKE: Students often assume that the composite function is always a function, without checking the properties of the individual functions. However, if the range of g is not a subset of the domain of f, the composite function may not be a function.