Question

does the graph represent a function and if so, why?
no, there is a limited number of ordered pairs on this graph.
yes, no two ordered pairs on this graph have the same first element.
no, there is no value for x = 0.
yes, no two ordered pairs on this graph have the same second element.

Explanation:

To determine if a graph represents a function, we use the vertical line test, which means checking if any x - value (first element of an ordered pair \((x,y)\)) is paired with more than one y - value.

Step 1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). In terms of ordered pairs \((x,y)\), no two ordered pairs should have the same first element (x - value) with different second elements (y - values).

Step 2: Analyze each option

• Option 1: The number of ordered pairs (whether limited or not) does not determine if a relation is a function. A function can have a finite or infinite number of ordered pairs. So this option is incorrect.
• Option 2: If no two ordered pairs on the graph have the same first element, that means each x - value is paired with only one y - value. This satisfies the definition of a function. Let's check the other options to be sure.
• Option 3: The graph does have a value for \(x = 0\) (we can see a point at \(x=0\) on the graph). Also, the presence or absence of a value at a single \(x\) - value does not determine if it's a function. So this option is incorrect.
• Option 4: A function allows for different x - values to have the same y - value (the second element). For example, \(y=x^{2}\) has \((2,4)\) and \((- 2,4)\), and it is a function. So the condition about the second element is not the defining condition for a function. This option is incorrect.

Answer:

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