太深奥了, 不懂。
我认为x, y的inverse关系, 应该是y=a/x。
In
mathematics, an
inverse function (or
anti-function[1]) is a
function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e.,
f(
x) =
y if and only if g(
y) =
x.
[2][3]
As an example, consider the real-valued function of a real variable given by
f(
x) = 5
x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case that means that we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by,
g ( y ) = y + 7 5 . {\displaystyle g
={\frac {y+7}{5}}.}
With
y = 5
x − 7 we have that
f(
x) =
y and
g(
y) =
x.
Not all functions have inverse functions. In order for a function
f:
X →
Y to have an inverse,
[nb 1] it must have the property that
for every y in Y there must be
one, and only one x in X so that
f(
x) =
y. This property ensures that a function
g:
Y →
X will exist having the necessary relationship with f.
Let f be a function whose
domain is the
set X, and whose
image (range) is the set Y. Then f is
invertible if there exists a function g with domain Y and image X, with the property:
f ( x ) = y ⇔ g ( y ) = x . {\displaystyle f(x)=y\,\,\Leftrightarrow \,\,g
=x.}
If f is invertible, the function g is
unique,
[4] which means that there is exactly one function g satisfying this property (no more, no less). That function g is then called
the inverse of f, and is usually denoted as
f −1.
[nb 2]
Stated otherwise, a function, considered as a
binary relation, has an inverse if and only if the
converse relation is a function on the range Y, in which case the converse relation is the inverse function.
[5]
Not all functions have an inverse. For a function to have an inverse, each element
y ∈
Y must correspond to no more than one
x ∈
X; a function f with this property is called one-to-one or an
injection. If
f −1 is to be a
function on Y, then each element
y ∈
Y must correspond to some
x ∈
X. Functions with this property are called
surjections. This property is satisfied by definition if Y is the image (range) of f, but may not hold in a more general context. To be invertible a function must be both an injection and a surjection. Such functions are called
bijections. The inverse of an injection
f:
X →
Y that is not a bijection, that is, a function that is not a surjection, is only a
partial function on Y, which means that for some
y ∈
Y,
f −1(
y) is undefined. If a function f is invertible, then both it and its inverse function
f−1 are bijections.
There is another convention used in the definition of functions. This can be referred to as the "set-theoretic" or "graph" definition using
ordered pairs in which a
codomain is never referred to.
[6] Under this convention all functions are surjections,
[nb 3] and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection.
[7] The two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function.
Example: Squaring and square root functions
The function
f: ℝ → [0,∞) given by
f(
x) =
x2 is not injective since each possible result
y (except 0) corresponds to two different starting points in X – one positive and one negative, and so this function is not invertible. With this type of function it is impossible to deduce an input from its output. Such a function is called non-
injective or, in some applications, information-losing.[
citation needed]
If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be
f: [0, ∞) → [0, ∞) with the same
rule as before, then the function is bijective and so, invertible.
[8] The inverse function here is called the
(positive) square root function.