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 $\mathbb {R} ^{2}$. 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 $\mathbb {R} ^{3}$ 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 $\mathbb {R} ^{n+1}$ 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.