Question

does the graph represent a function and if so, why?

graph of a line on a coordinate plane

○ 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 each input (first element of ordered pair, \(x\)-value) has exactly one output (second element, \(y\)-value). The first option states "no two ordered pairs on this graph have the same first element", which aligns with the function definition (each \(x\) maps to one \(y\)). The second option is incorrect because a function allows same second elements (e.g., \(y = x^2\) has symmetric points). The third option is wrong as having multiple ordered pairs doesn’t define a function. The fourth is incorrect as the number of ordered pairs (even infinite for lines) doesn’t determine if it’s a function, and the graph here is a line (linear function) which passes the vertical line test.

Answer:

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