Question

y² = x
two outputs for some inputs

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). In other words, for a relation to be a function, when we draw a vertical line through the graph, the vertical line should intersect the graph at most once (Vertical Line Test).

Step2: Apply the Vertical Line Test to \(y^{2}=x\)

For the equation \(y^{2}=x\), we can solve for \(y\) as \(y = \pm\sqrt{x}\) (for \(x\geq0\)). When we take a vertical line (for example, the yellow vertical line in the graph), it intersects the graph of \(y^{2}=x\) at two points (as shown in the graph: one point from \(y=\sqrt{x}\) and one point from \(y =-\sqrt{x}\)). This means that for some input \(x\) (the x - value corresponding to the vertical line), there are two different output \(y\) - values.

Step3: Conclusion based on the test

Since the graph of \(y^{2}=x\) fails the Vertical Line Test (a vertical line intersects the graph at more than one point), the relation \(y^{2}=x\) is not a function.

Answer:

The relation \(y^{2}=x\) is not a function because it fails the Vertical Line Test (a vertical line intersects its graph at two points for some \(x\) - values, meaning one input has two outputs).