Question

which of these is not a function?
(5, 3), (6, 4), (7, 3), (8, 4)
graph of a curve
$y = 3x^2$
graph of a parabola opening to the right

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). We can use the vertical line test for graphs: if any vertical line intersects the graph at more than one point, it is not a function. For ordered pairs, each x - value must be unique or map to only one y - value.

Step2: Analyze the first set of ordered pairs

The ordered pairs are \((5,3)\), \((6,4)\), \((7,3)\), \((8,4)\). The x - values are \(5\), \(6\), \(7\), \(8\) (all unique), so each x has one y. This is a function.

Step3: Analyze the second graph (the curve)

Using the vertical line test, any vertical line will intersect this curve at most once. So this graph represents a function.

Step4: Analyze the equation \(y = 3x^{2}\)

For any real number \(x\), plugging in \(x\) into the equation gives exactly one \(y\) - value. So this is a function.

Step5: Analyze the fourth graph (the curve)

Using the vertical line test, if we draw a vertical line (e.g., \(x = 1\)), it will intersect the graph at two points (one in the upper half and one in the lower half). So this graph does not pass the vertical line test and is not a function.

Answer:

The graph (the last one with the curve that is symmetric about the x - axis, passing through the origin with two branches) is NOT a function. (In terms of the options, the last option with the graph that fails the vertical line test)