Vertical line test

Statement

For a function of one variable

Forward direction: Suppose $f$ is a real-valued function of one variable $x$ . The vertical line test says that any vertical line, i.e., any line of the form $x = x_{0}$ , intersects the graph $y = f (x)$ of the function at at most one point. Further, there is a point of intersection if and only if $x_{0}$ is in the domain of the function. Otherwise, there is no point of intersection.

Reverse direction: Suppose $A$ is any subset of $R^{2}$ . Then, $A$ occurs as the graph of a function if and only if the intersection of $A$ with every vertical line has size at most one. Further, the function is uniquely determined by $A$ , and is given as the function $f$ where:

A point $x_{0}$ is in the domain of $f$ if and only if the line $x = x_{0}$ intersects the set $A$ .
For such a point $x_{0}$ , $f (x_{0})$ is defined as the $y$ -coordinate of the intersection.

For a function of two variables

Forward direction: Suppose $f$ is a real-valued function of two variables $x, y$ . Imagine that we are in a three-dimensional space with coordinates $x, y, z$ .

The vertical line test says that any line parallel to the $z$ -axis, i.e., any line of the form $x = x_{0}, y = y_{0}$ , intersects the graph $z = f (x, y)$ of the function at at most one point. Further, there is a point of intersection if and only if $(x_{0}, y_{0})$ is in the domain of the function. Otherwise, there is no point of intersection.

Reverse direction: Suppose $A$ is any subset of the three-dimensional space $R^{3}$ with coordinates $x, y, z$ . Then, $A$ occurs as the graph of a function if and only if the intersection of $A$ with every line parallel to the $z$ -axis has size at most one. Further, the function is uniquely determined by $A$ , and is given as the function $f$ where:

A point $(x_{0}, y_{0})$ is in the domain of $f$ if and only if the line $x = x_{0}, y = y_{0}$ intersects the set $A$ .
For such a point $(x_{0}, y_{0})$ , $f (x_{0}, y_{0})$ is defined as the $z$ -coordinate of the intersection.

For a function of multiple variables

Forward direction: Suppose $f$ is a real-valued function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . Imagine that we are in $(n + 1)$ -dimensional space with coordinates $x_{1}, x_{2}, \dots, x_{n}$ .

The vertical line test says that any line parallel to the $x_{n + 1}$ -axis, i.e., any line of the form $x_{1} = a_{1}, x_{2} = a_{2}, \dots, x_{n} = a_{n}$ , intersects the graph $x_{n + 1} = f (x_{1}, x_{2}, \dots, x_{n})$ of the function at at most one point. Further, there is a point of intersection if and only if $(a_{1}, a_{2}, \dots, a_{n})$ is in the domain of the function. Otherwise, there is no point of intersection.

Reverse direction: Suppose $A$ is any subset of the $(n + 1)$ -dimensional space $R^{n + 1}$ with coordinates $x_{1}, x_{2}, \dots, x_{n}, x_{n + 1}$ . Then, $A$ occurs as the graph of a function if and only if the intersection of $A$ with every line parallel to the $x_{n + 1}$ -axis has size at most one. Further, the function is uniquely determined by $A$ , and is given as the function $f$ where:

A point $(a_{1}, a_{2}, \dots, a_{n})$ is in the domain of $f$ if and only if the line $x_{1} = a_{1}, x_{2} = a_{2}, \dots, x_{n} = a_{n}$ intersects the set $A$ .
For such a point $(a_{1}, a_{2}, \dots, a_{n})$ , the value $f (a_{1}, a_{2}, \dots, a_{n})$ is defined as the $x_{n + 1}$ -coordinate of the intersection.