Problem 69 - Thomas Calculus

Question

For a curve to be symmetric about the \(x\) -axis, the point \((x, y)\) must lie on the curve if and only if the point \((x,-y)\) lies on the curve. Explain why a curve that is symmetric about the \(x\) -axis is not the graph of a function, unless the function is \(y=0 .\)

Step-by-Step Solution

Answer

A curve symmetric about the x-axis cannot be a function (except for \(y=0\)) because it contradicts the requirement of a unique y-value for each x-value.

Step 1: Understanding Symmetry about the x-axis

A curve is symmetric about the x-axis if for every point \((x, y)\) on the curve, the point \((x, -y)\) is also on the curve. This means if you flip the curve over the x-axis, it remains unchanged. We need to explain why such symmetry is not possible for a function, except for the function \(y = 0\).

Step 2: Definition of a Function

By definition, a function is a relation where each input \(x\) has exactly one output \(y\). For a curve to be a function of \(x\), at any given \(x\)-value, there must be only one corresponding \(y\)-value.

Step 3: Conflict with Function Definition

If a curve is symmetric about the x-axis, for any \((x, y)\), there is also a point \((x, -y)\). This means for an x-value, we have two possible y-values, \(y\) and \(-y\), which conflicts with the definition of a function. Thus, such curves cannot represent functions.

Step 4: Exception with the Line y=0

The only exception is when \(y=0\). In this case, the points \((x, 0)\) and \((x, -0)\) are the same, meaning there is only one y-value (which is 0) for each x-value. Therefore, the line \(y=0\) (the x-axis itself) is the only curve symmetric about the x-axis that is a function.

Key Concepts

x-axis symmetry

In mathematical terms, when we say a curve is symmetric about the x-axis, it means if the point \((x, y)\) is on the curve, the point \((x, -y)\) must also be on the curve. This creates a mirror effect where any part of the curve above the x-axis has an exact replica below it. Such symmetry implies that if you visually "flip" the curve over the x-axis, its appearance remains unchanged.
However, this characteristic introduces a peculiar complication when considering functions. A function, by definition, associates each x-value with one and only one y-value. With x-axis symmetry, each x-value would have two y-values (one positive and one negative), which contradicts the definition of a function. This is why curves with x-axis symmetry cannot, in general, represent functions.

function definition

A function in mathematics is a special kind of relation. What makes it special is the requirement that each input value \(x\) must map to one and only one output value \(y\). This means that for any given x-value, there can only be one corresponding y-value, ensuring a unique pathway from inputs to outputs.
Take note of the vertical line test: if a vertical line can intersect a curve at more than one point, the curve is not the graph of a function. With x-axis symmetry, a vertical line can hit multiple corresponding y-values for the same x-value, creating a conflict. This rule keeps in check the definition of a function, guaranteeing clarity and consistency in mapping inputs to outputs.

graph of a function

The graph of a function is a visual representation of all the possible points \((x, y)\) derived from the function's rule. By plotting each pair just once, it forms a unique path or line on the coordinate plane, reflecting the function's behavior.
For example, the graph of \(y=x^2\) curves upward as every x-value maps to one distinct y-value using the function's rule. However, a curve symmetric about the x-axis breaks this pattern. It would mean multiple y-values correspond to the same x-value, thus failing the vertical line test, as it's not possible for a function.
A notable exception exists: the horizontal line where \(y=0\). This "graph" passes the vertical line test perfectly, as for every x-value, the output y is always 0. No contradiction arises, making it the only curve showing x-axis symmetry that fits a function's criteria.