Question

x = {(2, 4), (3, 4), (4, 4), (5, 4)} is x a function and why?
○ yes, no two ordered pairs in this list has the same first element.
○ no, each set of ordered pairs in this list has the same second element.
○ no, there is a limited number of ordered pairs in this list.
○ yes, there is more than one ordered pair in this list.

Explanation:

To determine if a relation (set of ordered pairs) is a function, the key rule is that no two ordered pairs can have the same first element (input) with different second elements (outputs). In the set \( x = \{(2, 4), (3, 4), (4, 4), (5, 4)\} \), we check the first elements of each ordered pair: 2, 3, 4, and 5. None of these first elements are repeated. The fact that the second element (4) is the same for all pairs does not violate the function definition, as multiple inputs can map to the same output (this is called a constant function).

Now let's analyze each option:

• Option 1: "Yes, no two ordered pairs in this list has the same first element." This matches the function definition because the first elements (2, 3, 4, 5) are all unique.
• Option 2: "No, each set of ordered pairs in this list has the same second element." A function allows multiple inputs to have the same output, so this reasoning is incorrect.
• Option 3: "No, there is a limited number of ordered pairs in this list." The number of ordered pairs (whether limited or not) has no bearing on whether a relation is a function.
• Option 4: "Yes, there is more than one ordered pair in this list." The number of ordered pairs does not determine if a relation is a function; the key is the uniqueness of first elements.

Answer:

A. Yes, no two ordered pairs in this list has the same first element.