Explanation:
Function (mathematics)
A function, in mathematics, associates one
quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as
the output. A function assigns exactly
one output to each input. The argument and the value may be real numbers, but
they can also be elements from any given set. An example of a function is f(x) = 2x, a
function which associates with every number the number twice as large. Thus 5
is associated with 10, and this is written f(5) = 10.
http://en.wikipedia.org/wiki/Function_(mathematics)
Inverse
function
A
function ƒ and its inverse ƒ^{-1}. Because ƒ maps a to
3, the inverse ƒ^{-1} maps 3 back to a.
In mathematics, an inverse function is a function that undoes another function: If an
input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ(x)=y,
and g(y)=x.
More directly, g(ƒ(x))=x,
meaning g(x)
composed with ƒ(x) leaves x unchanged.
A function ƒ that has an inverse is called invertible; the inverse
function is then uniquely determined by ƒ and is denoted by ƒ^{−1} (read f inverse, not to be confused
with exponentiation).
A relation can be determined to have an inverse if it is a
one-to-one function.
(Our solved
example in mathguru.com uses this concept)
Definitions
If ƒ
maps X to Y, then ƒ^{-1} maps Y back
to X.
Instead of considering
the inverses for individual inputs and outputs, one can think of the function
as sending the whole set of inputs, the domain, to a set of outputs, the range.
Let ƒ be a function whose domain is the set X, and whose range is the set Y. Then ƒ is invertible if there exists a function g with
domain Y and range X, with
the property:
If ƒ is invertible, the function g is unique; in other words, there can
be at most one function g satisfying this property. That
function g is then called the inverse of ƒ, denoted by ƒ^{−1}.
Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse
relation is the inverse function.
Not all functions have an inverse. For this rule to be applicable,
each element y ∈ Y must correspond to no more than one x ∈ X;
a function ƒ with this property is called one-to-one, or
information-preserving, or an injection.
Inverses and composition
If ƒ is an invertible function with domain X and range Y, then
This statement is
equivalent to the first of the above-given definitions of the inverse, and it
becomes equivalent to the second definition if Y coincides
with the codomain of ƒ. Using the composition of functions we
can rewrite this statement as follows:
where id_{X} is
the identity function on the set X; that is, the function that
leaves X unchanged. (Our
solved example in mathguru.com uses this concept)
In category theory,
this statement is used as the definition of an inverse morphism.
If we think of composition as a kind of multiplication of
functions, this identity says that the inverse of a function is analogous to a multiplicative inverse.
Properties
Uniqueness
If an inverse function exists for a given function ƒ, it is
unique: it must be the inverse
relation.
Symmetry
There is symmetry between a function and its inverse.
Specifically, if ƒ is an invertible function with domain X and range Y, then its inverse ƒ^{−1} has domain Y and range X, and the inverse of ƒ^{−1} is the original function ƒ. In symbols,
for ƒ a function with domain X and range Y, and g a function with domain Y and range X:
This statement is an obvious consequence
of the deduction that for ƒ to be invertible it must be injective (first
definition of the inverse) or bijective (second definition). The property of
symmetry can be concisely expressed by the following formula:
The
inverse of g o ƒ is ƒ^{-1} o g^{-1}.
The inverse of a composition of functions
is given by the formula
Notice that the order of g and f have
been reversed; to undo f followed by g, we must
first undo g and then undo f.
For example, let f(x)
= 3x and let g(x) = x + 5. Then
the composition g o f is the function that
first multiplies by three and then adds five:
To reverse this process, we must first
subtract five, and then divide by three:
This is the composition (f^{-1} o g^{-1})
(y).
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Inverse_function
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