1.3 Composition of Functions and Invertible Function > No. 7

Find the inverse of the given function.

Post to:

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:

$Description: f(x) = y\,\,\text{if and only if}\,\,g(y) = x\text{.}\,\!$

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

$Description: f^{-1}\left( \, f(x) \, \right) = x\text{, for every }x \in X\text{.}$

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:

$Description: f^{-1} \circ f = \mathrm{id}_X\text{,}$

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:

$Description: \begin{align} &\text{If } &g \circ f = \mathrm{id}_X\text{,} \\ &\text{then } &f \circ g = \mathrm{id}_Y\text{.} \end{align}$

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:

$Description: \left(f^{-1}\right)^{-1} = f . \,\!$

The inverse of g o ƒ is ƒ^-1 o g^-1.

The inverse of a composition of functions is given by the formula

$Description: (g \circ f)^{-1} = f^{-1} \circ g^{-1}$

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:

$Description: (g \circ f)(x) = 3x + 5$

To reverse this process, we must first subtract five, and then divide by three:

$Description: (g \circ f)^{-1}(y) = \tfrac13(y - 5)$

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

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.