From MathWorld, a function is an object
such that every
is uniquely associated with an object
. Stated more pedantically, if
and
, then
. More colloquially, in the graphs that ordinarily appear in secondary school, every
coordinate of the graph is associated with a unique
coordinate.
For this reason, the figure below (taken from http://en.wikipedia.org/wiki/Vertical_line_test) is not a function. The three points share a common coordinate but have different
coordinates. In school, we usually teach students to distinguish functions from non-functions by the Vertical Line Test.
It is possible for a function to be a function but not have an inverse. Also from MathWorld, a function is said to be an injection (or, in the lingo that I learned as a student, one-to-one) if, whenever
, it must be the case that
. Equivalently,
implies
. In other words,
is an injection if it maps distinct objects to distinct objects.
The following image (taken from http://en.wikipedia.org/wiki/Horizontal_line_test) illustrates a function that is not injective (or, more accurately, is not injective when using all of the function’s domain). The horizontal line intersects the graph of the function at three distinct points with three different intercepts which are associated with the same
-coordinate.
By ensuring that the range of is restricted to the values that are actually attained by
, the function
may be considered as bijective and hence has an inverse function. The inverse function
is logically defined as
In this way, and
for all
in the domain of
and all
in the range of
. Becaise
is on the graph of
if and only if
is on the graph of
, the graph of
may be obtained by reflecting the graph of
through the line
. Stated another way, to ensure that
is a function that satisfies the horizontal line test, it must be the case (when looking at the reflection through
that the original function satisfies the horizontal line test.
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