Can a function have multiple relationships?

📌 CONCEPT: A function can have multiple relations, but a relation can be satisfied by a function if it satisfies the condition that each element in the domain is related to exactly one element in the codomain.

📐 RULE / FORMULA: For a relation R to be satisfied by a function f(x), the function must be one-to-one, meaning that each element in the domain is related to exactly one element in the codomain.

💡 WORKED EXAMPLE: Consider the relation R = {(1, 3), (2, 5), (3, 7)}. We can define a function f(x) = 2x + 1 that satisfies R. For instance, f(1) = 3, f(2) = 5, and f(3) = 7. The graph of f(x) would be a straight line passing through the points (1, 3), (2, 5), and (3, 7).

⚠️ COMMON MISTAKE: Students often mistakenly believe that a function can have multiple relations, but this is not true. A function can have multiple relations, but each relation must be satisfied by the function.