Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 55-56: Explain why the graph of the equation is not the graph of a function. $$ x=y^{2} $$

Step-by-Step Solution

Answer

The graph fails the vertical line test, thus it is not a function.

Step 1: Understand the Definition of a Function

A relation is a function if every input (usually represented by \(x\)) corresponds to exactly one output (usually represented by \(y\)). This means for a graph to represent a function, a vertical line should intersect the graph at most once.

Step 2: Rewrite the Given Equation

The equation given is \(x = y^2\). Write it in terms of \(y\): \(y = \pm \sqrt{x}\). This means for each value of \(x\), there are possibly two values of \(y\).

Step 3: Apply the Vertical Line Test

The vertical line test states that if a vertical line intersects the graph of an equation more than once for any value of \(x\), then the graph does not represent a function. In the equation \(x = y^2\), for any \(x > 0\), the vertical line will intersect points \((x, \sqrt{x})\) and \((x, -\sqrt{x})\).

Step 4: Conclusion

Since there exist values of \(x\) for which vertical lines intersect the graph at more than one \(y\)-value, the given graph does not satisfy the definition of a function.

Key Concepts

Vertical Line Test

The vertical line test is a simple yet powerful tool to determine if a graph represents a function. The essence of this test is straightforward: if you can draw a vertical line anywhere on the graph, and it crosses the graph in more than one place simultaneously, it means the graph does **not** represent a function.

The reasoning behind this is rooted in the definition of a function. A function associates each input with exactly one output. If a vertical line crosses the graph more than once, there are multiple outputs for a single input, which violates the property of a function.

For the equation in question, \(x = y^2\), we see that vertical lines intersect the graph at two points for \(x > 0\). This shows that the graph fails the vertical line test and is therefore not a graph of a function.

Relation

A relation in mathematics is a set of ordered pairs. It represents how two sets of information are connected. In simpler terms, if you have two related things, like inputs and outputs, their interaction forms a **relation**.

For example, in the equation \(x = y^2\), every value of \(x\) has a corresponding \(y\) or possibly two \(y\) values, as seen when rewritten as \(y = \pm \sqrt{x}\). This confirms that the equation describes a relation because it lists pairs of \((x, y)\) that relate inputs to outputs.

However, not all relations are functions. A relation becomes a function only if each input has one and only one output. Since the equation \(x = y^2\) relates one \(x\) to two potential \(y\) outcomes, it remains a relation but not a function.

Graph of an Equation

The graph of an equation is a visual representation of all solutions of the equation on a coordinate plane. It shows where the equation holds true for the values of \(x\) and \(y\).

To sketch the graph of the equation \(x = y^2\), we see it describes a parabola that opens sideways, plotted on the \(xy\)-plane. When plotted, it will intersect the vertical axis at multiple points indicating more outputs for a single input.

Graphing helps us understand the nature of the equation and visually confirms characteristics such as whether the relation is a function or not. In this case, applying the vertical line test on this graph clearly shows it is not a function, as vertical lines can intersect it in more than one point for some \(x\) values.