Math Problem Statement

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

Solution

A curve that is symmetric about the x-axis is not the graph of a function unless the function is y=0y = 0 due to the definition of a function in mathematics.

Explanation:

  1. Definition of a Function: A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In terms of a graph, this means that for each xx-value, there can be only one corresponding yy-value.

  2. Symmetry About the x-Axis: If a curve is symmetric about the x-axis, it means that for every point (x,y)(x, y) on the curve, there is a corresponding point (x,y)(x, -y) also on the curve. This symmetry implies that the curve has two different yy-values (one positive and one negative) for the same xx-value, except at points where y=0y = 0.

  3. Violating the Function Definition: Since a function must assign exactly one yy-value to each xx-value, a curve that is symmetric about the x-axis generally does not meet this requirement because it assigns two yy-values to a single xx-value. The only exception is when y=0y = 0, as this does not create a conflict (both the point (x,0)(x, 0) and its symmetric counterpart (x,0)(x, 0) have the same yy-value).

  4. Conclusion: Therefore, a curve that is symmetric about the x-axis cannot be the graph of a function unless the function is y=0y = 0, which is the only scenario where each xx-value corresponds to a single yy-value.

Would you like more details on this concept, or do you have any questions?

Related Questions:

  1. What are some examples of curves that are symmetric about the x-axis?
  2. How does symmetry about the y-axis affect whether a curve is a function?
  3. Can you explain the Vertical Line Test and how it determines whether a curve is a function?
  4. What is the significance of the line y=0y = 0 in mathematical functions?
  5. How does symmetry about the origin differ from symmetry about the x-axis?

Tip:

Always use the Vertical Line Test on a graph to quickly determine if it represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

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Math Problem Analysis

Mathematical Concepts

Functions
Symmetry
Graph Theory

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12