Question

does the graph represent a function and if so, why?
yes, no two ordered pairs on this graph have the same first element.
yes, no two ordered pairs on this graph have the same second element.
yes, there is more than one ordered pair on this graph.
no, there is a limited number of ordered pairs on this graph.

Explanation:

To determine if a graph represents a function, we use the vertical line test, which means checking if no two ordered pairs have the same first element (x - value). Let's analyze each option:

• Option 1: A function is defined as a relation where each input (first element of an ordered pair, \(x\) - value) has exactly one output (second element, \(y\) - value). If no two ordered pairs have the same first element, then each \(x\) is mapped to only one \(y\), so the graph represents a function.
• Option 2: The definition of a function does not require that no two ordered pairs have the same second element ( \(y\) - value). A function can have different \(x\) - values mapping to the same \(y\) - value (e.g., \(y = x^{2}\), where \(x = 2\) and \(x=- 2\) both map to \(y = 4\)). So this is not the correct reason.
• Option 3: The number of ordered pairs (whether more than one or not) is not the defining characteristic of a function. A function can have one or more ordered pairs as long as the mapping from \(x\) to \(y\) is unique for each \(x\). So this is incorrect.
• Option 4: The number of ordered pairs (whether limited or not) has nothing to do with whether a graph represents a function. A function can be defined with a finite or infinite number of ordered pairs as long as the vertical line test is satisfied. So this is incorrect.

Answer:

Yes, no two ordered pairs on this graph have the same first element.