Chapter 0.2, Problem 36E

To determine

To explain:

Why a curve that is symmetric about the $x$ axis is not the graph of a function, unless the function is $y=0$ .

Answer to Problem 36E

According to the given definition of curves symmetric about the $x$ axis if any point not on the $x$ axis $\left(a,b\right);b\ne 0$ lies on the curve then the point $\left(a,-b\right)$ must also lie on the curve. It follows that the curve passes through two distinct points $\left(a,b\right)$ and $\left(a,-b\right)$ that have the same $x$ coordinate, and thus it cannot be the graph of a function.

The only graph of a function which qualifies to be called symmetric about the $x$ axis is a curve all whose points lie on the $x$ axis, namely the line represented by the function $y=0$ .

Explanation of Solution

Given:

All points on the $x$ axis have the $y$ coordinate equal to zero.

For a curve to be symmetric about the $x$ axis, the point $\left(x,y\right)$ must lie on the curve if and only if the point $\left(x,-y\right)$ lies on the curve.

Concepts Used:

For a curve on $xy$ plane to be considered the graph of a function, no two distinct points on it must have the same $x$ coordinate.

Description:

According to the given definition of curves symmetric about the $x$ axis if any point not on the $x$ axis $\left(a,b\right);b\ne 0$ lies on the curve then the point $\left(a,-b\right)$ must also lie on the curve. It follows that the curve passes through two distinct points $\left(a,b\right)$ and $\left(a,-b\right)$ that have the same $x$ coordinate, and thus it cannot be the graph of a function.

The only graph of a function which qualifies to be called symmetric about the $x$ axis is a curve all whose points lie on the $x$ axis, namely the line represented by the function $y=0$ .