Vipul: Created page with "==Statement== ===For a function of one variable=== '''Forward direction''': Suppose $f$ is a real-valued function of one variable $x$ . The '''vertical ..."

2012-05-08T21:13:30Z

Created page with "==Statement== ===For a function of one variable=== '''Forward direction''': Suppose <math>f</math> is a real-valued function of one variable <math>x</math>. The '''vertical ..."

New page

==Statement==

===For a function of one variable===

'''Forward direction''': Suppose <math>f</math> is a real-valued function of one variable <math>x</math>. The '''vertical line test''' says that any vertical line, i.e., any line of the form <math>x = x_0</math>, intersects the [[graph]] <math>y = f(x)</math> of the function at at most one point. Further, there is a point of intersection if and only if <math>x_0</math> is in the [[domain]] of the function. Otherwise, there is no point of intersection.

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

* A point <math>x_0</math> is in the domain of <math>f</math> if and only if the line <math>x = x_0</math> intersects the set <math>A</math>.
* For such a point <math>x_0</math>, <math>f(x_0)</math> is defined as the <math>y</math>-coordinate of the intersection.

===For a function of two variables===

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

The '''vertical line test''' says that any line parallel to the <math>z</math>-axis, i.e., any line of the form <math>x = x_0, y = y_0</math>, intersects the [[graph]] <math>z = f(x,y)</math> of the function at at most one point. Further, there is a point of intersection if and only if <math>(x_0,y_0)</math> is in the [[domain]] of the function. Otherwise, there is no point of intersection.

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

* A point <math>(x_0,y_0)</math> is in the domain of <math>f</math> if and only if the line <math>x = x_0, y = y_0</math> intersects the set <math>A</math>.
* For such a point <math>(x_0,y_0)</math>, <math>f(x_0,y_0)</math> is defined as the <math>z</math>-coordinate of the intersection.

===For a function of multiple variables===

'''Forward direction''': Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Imagine that we are in <math>(n+1)</math>-dimensional space with coordinates <math>x_1,x_2,\dots,x_n</math>.

The '''vertical line test''' says that any line parallel to the <math>x_{n+1}</math>-axis, i.e., any line of the form <math>x_1 = a_1, x_2 = a_2, \dots, x_n = a_n</math>, intersects the [[graph]] <math>x_{n+1} = f(x_1,x_2,\dots,x_n)</math> of the function at at most one point. Further, there is a point of intersection if and only if <math>(a_1,a_2,\dots,a_n)</math> is in the [[domain]] of the function. Otherwise, there is no point of intersection.

'''Reverse direction''': Suppose <math>A</math> is any subset of the <math>(n+1)</math>-dimensional space <math>\R^{n+1}</math> with coordinates <math>x_1,x_2,\dots,x_n,x_{n+1}</math>. Then, <math>A</math> occurs as the graph of a function if and only if the intersection of <math>A</math> with every line parallel to the <math>x_{n+1}</math>-axis has size at most one. Further, the function is uniquely determined by <math>A</math>, and is given as the function <math>f</math> where:

* A point <math>(a_1,a_2,\dots,a_n)</math> is in the domain of <math>f</math> if and only if the line <math>x_1 = a_1, x_2 = a_2, \dots, x_n =a_n</math> intersects the set <math>A</math>.
* For such a point <math>(a_1,a_2,\dots,a_n)</math>, the value <math>f(a_1,a_2,\dots,a_n)</matH> is defined as the <math>x_{n+1}</math>-coordinate of the intersection.

Vertical line test - Revision history

Vipul: Created page with "==Statement== ===For a function of one variable=== '''Forward direction''': Suppose f is a real-valued function of one variable x. The '''vertical ..."

Vipul: Created page with "==Statement== ===For a function of one variable=== '''Forward direction''': Suppose $f$ is a real-valued function of one variable $x$ . The '''vertical ..."