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Chapter 1: Problem 69

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 \(y=0.\)

Short Answer

Expert verified
Symmetry about the \(x\)-axis means \((x, y)\) and \((x, -y)\) are on the curve, failing the vertical line test unless \(y = 0\).

Step by step solution

01

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 that for the point \((x, y)\), its mirror image across the \(x\)-axis also lies on the curve.
02

Defining the Graph of a Function

A function has a unique output \(y\) for every input \(x\). Formally, a relation \(y = f(x)\) is a function if each \(x\) corresponds to exactly one \(y\). For graphs, this means that a vertical line can intersect the graph at most once.
03

Applying the Vertical Line Test

The vertical line test states that if a vertical line intersects a graph at more than one point, the graph is not of a function. For symmetry about the \(x\)-axis, points \((x, y)\) and \((x, -y)\) would both be on the graph, causing a vertical line through \(x\) to intersect the graph twice unless each \(y\) is zero, which means \((x, y) = (x, 0)\).
04

Identifying the Exception: y = 0

The only scenario where a curve symmetric about the \(x\)-axis is a function is when all points satisfy \(y = 0\). This is because then, \((x, y)\) and \((x, -y)\) are the same point \((x, 0)\). Hence, the graph is just the \(x\)-axis, which satisfies the condition of a function with \(y = f(x) = 0\) for all \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Graph
A function graph is a visual representation of a function, where each input value (x) is linked to a single output value (y). This relationship ensures clarity, as the graph helps to understand the behavior of the function over its domain. To plot a function, one can create a set of ordered pairs (x, y), where 'x' is the input and 'y' is the corresponding output.
  • The graph provides insights into key points like intercepts, turning points, and asymptotes.
  • Functions can be linear, polynomial, exponential, or trigonometric, among others, each having distinct graph shapes.
It's crucial for the graph of a function to visually demonstrate the unique pairing of x values to their respective y values.
Vertical Line Test
The vertical line test is a simple and effective way to determine if a graph represents a function. The idea is straightforward: draw a vertical line anywhere on the graph.
  • If the line touches the graph at only one point, the graph is of a function.
  • If it touches the graph at more than one point, it fails the test, indicating the graph is not of a function.
For example, consider the symmetry about the x-axis: for points (x, y) and (x, -y), a vertical line through any x will intersect the graph twice unless y is zero everywhere. Thus, only the line y = 0 on the x-axis would pass the vertical line test for a graph symmetric about the x-axis.
Symmetric Curves
Symmetric curves exhibit reflective qualities about a specific axis or point. For a curve to be symmetric about the x-axis, whenever the point (x, y) is on a curve, its corresponding reflected point (x, -y) must also be present. This particular symmetry might lead to issues with the vertical line test, as previously discussed.
  • An x-axis symmetry implies that there are mirrored points directly below or above a line.
  • This kind of symmetry makes these curves fail to represent functions鈥攅xcept in trivial cases.
Symmetric curves are common in nature and art, playing a significant role in aesthetic and structural design. But as far as functions are concerned, their presence indicates potential complexity or limitations in defining simple relationships.
Unique Output of a Function
Uniqueness in a function is critical. By definition, each input x in a function corresponds to exactly one output y. This 'one-to-one' relationship is what differentiates functions from other types of relations that might allow multiple outputs for a single input.
  • The unique pairing guarantees consistency across the domain of the function.
  • It simplifies calculations and predictions, offering reliable analysis in mathematical modeling.
When a curve is symmetric about the x-axis, it disrupts this unique pairing鈥攗nless every y is 0, making it impossible for such a symmetric curve to satisfy the function definition. In essence, maintaining a function's unique output is fundamental to preserving its integrity and applicability.

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