Vertical line test
Encyclopedia
In mathematics, the vertical line test is a test to determine if a curve is a relation
or graph
of a function when the function's domain and Range
correspond to the x and y axes of the Cartesian coordinate system
. As a relation or graph of a function can only have one output for each unique input, such a Cartesian representation of the function can have at most a single y value for each x value. Thus, a vertical line drawn at any x position on the graph of a function will intersect the graph at most once.
To use the vertical line test graphically, take a ruler or other "vertical line" and move it from one end of the x-axis to the other while keeping it parallel to the y-axis. If the graph intersects the ruler or vertical line more than once at any given value of x, the graph is not a Euclidean function. However, if the graph intersects the vertical line no more than once at any given point, it is a function. For example, any line will be a function (other than a vertical one). However, the sideways parabola is not because a vertical line will hit the parabola twice.
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
or graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of a function when the function's domain and Range
Range
-In mathematics:* Range , the set of all output values produced by a function* Interval , also called a range, a set of real numbers that includes all numbers between any two numbers in the set...
correspond to the x and y axes of the Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
. As a relation or graph of a function can only have one output for each unique input, such a Cartesian representation of the function can have at most a single y value for each x value. Thus, a vertical line drawn at any x position on the graph of a function will intersect the graph at most once.
To use the vertical line test graphically, take a ruler or other "vertical line" and move it from one end of the x-axis to the other while keeping it parallel to the y-axis. If the graph intersects the ruler or vertical line more than once at any given value of x, the graph is not a Euclidean function. However, if the graph intersects the vertical line no more than once at any given point, it is a function. For example, any line will be a function (other than a vertical one). However, the sideways parabola is not because a vertical line will hit the parabola twice.