# 该死的英文， 总是让我有挫败感

#### sabre的马甲

##### 知名园友

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) = 5x − 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 = 5x − 7 we have that f(x) = y and g(y) = x.

Not all functions have inverse functions. In order for a function f: XY 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: YX 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 yY must correspond to no more than one xX; 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 yY must correspond to some xX. 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: XY 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 yY, 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.

#### shw019

##### 知名园友

reverse是反向的，那reversed就是被反转了的意思。

#### shw019

##### 知名园友

sleight s-l-eigh-t
slight s-ligh-t

sleigh s-l-eigh
eight eigh-t

#### bloodwolfmoon

##### 活跃园友

You can only fool me once

#### bloodwolfmoon

##### 活跃园友
dictatorship

Democracy 的反义词 Totalitarianism 最接近

#### bloodwolfmoon

##### 活跃园友
gumdrops是双关语， 医学上是牙床子脱落了。
I don't think so

#### sabre的马甲

##### 知名园友

Democracy 的反义词 Totalitarianism 最接近

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