Chapter 0.2, Problem 35E

To determine

To explain:

Why the given assertion known as Vertical Line Test is true.

Answer to Problem 35E

If a vertical line intersects a given curve at more than one points then there are at least two points on the curve which have the same $x$ coordinate and thus the curve cannot be the graph of the function. If there can be no vertical line which intersects the given curve at more than one point then there are no two distinct points on the curve which have the same $x$ coordinate and thus the curve represents the graph of a function.

Explanation of Solution

Given:

The vertical line test to determine whether a curve is the graph of a function states: if every vertical line in the $xy$ plane intersects given curve at at most one point, then the curve is the graph of a function.

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.

Vertical lines on the $xy$ plane have equations of the form $x=a$ where $a$ is a constant. In other word, all the points on the same vertical line have the same $x$ coordinate.

Description:

Recall that 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 and that vertical lines on the $xy$ plane have equations of the form $x=a$ where $a$ is a constant. In other word, all the points on the same vertical line have the same $x$ coordinate.

If a vertical line intersects a given curve at more than one points then there are at least two points on the curve which have the same $x$ coordinate and thus the curve cannot be the graph of the function. If there can be no vertical line which intersects the given curve at more than one point then there are no two distinct points on the curve which have the same $x$ coordinate and thus the curve represents the graph of a function.