Editing Inverse function (section)

== Examples ==

===Squaring and square root functions===
The function {{math|''f'': '''R''' → [0,∞)}} given by {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} is not injective because <math>(-x)^2=x^2</math> for all <math>x\in\R</math>. Therefore, {{Mvar|f}} is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function <math>f\colon [0,\infty)\to [0,\infty);\ x\mapsto x^2</math> with the same ''rule'' as before, then the function is bijective and so, invertible.<ref>{{harvnb|Lay|2006|loc=p. 69, Example 7.24}}</ref> The inverse function here is called the ''(positive) square root function'' and is denoted by <math>x\mapsto\sqrt x</math>.
<!-- Repetitive. To be held for a short time until refactor is finished. 
===Inverses in higher mathematics===
The definition given above is commonly adopted in [[set theory]] and [[calculus]]. In higher mathematics, the notation
:<math>f\colon X \to Y </math>
means "{{mvar|f}} is a function mapping elements of a set {{mvar|X}} to elements of a set {{mvar|Y }}". The source, {{mvar|X}}, is called the domain of {{mvar|f}}, and the target, {{mvar|Y}}, is called the [[codomain]]. The codomain contains the range of {{mvar|f}} as a [[subset]], and is part of the definition of {{mvar|f}}.<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=p. 179}}</ref>

When using codomains, the inverse of a function {{math| ''f'': ''X'' → ''Y''}} is required to have domain {{mvar|Y}} and codomain {{mvar|X}}. For the inverse to be defined on all of {{mvar|Y}}, every element of {{mvar|Y}} must lie in the range of the function {{mvar|f}}. A function with this property is called ''onto'' or ''[[Surjective function|surjective]]''. Thus, a function with a codomain is invertible if and only if it is both ''[[Injective function|injective]]'' (one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a [[bijection]], and has the property that every element {{math| ''y'' ∈ ''Y''}} corresponds to exactly one element {{math| ''x'' ∈ ''X''}}.
-->

=== Standard inverse functions ===
The following table shows several standard functions and their inverses:
{| class="wikitable" align="center"
|+Inverse arithmetic functions
|-
!scope="col" align="center" | Function {{math|''f''(''x'')}}
!scope="col" align="center" | Inverse {{math|''f''<sup> −1</sup>(''y'')}}
!scope="col" align="center" | Notes
|-
| align="center" | {{math|''x'' [[addition|+]] ''a''}}
| align="center" | {{math|''y'' [[subtraction|−]] ''a''}}
|
|-
| align="center" | {{math|''a'' − ''x''}}
| align="center" | {{math|''a'' − ''y''}}
|
|-
| align="center" | {{math|[[multiplication|''mx'']]}}
| align="center" | {{sfrac|{{mvar|y}}|{{mvar|m}}}}
| {{math|''m'' ≠ 0}}
|-
| align="center" | {{sfrac|1|{{mvar|x}}}} (i.e. {{math|''x''<sup>−1</sup>}})
| align="center" | {{sfrac|1|{{mvar|y}}}} (i.e. {{math|''y''<sup>−1</sup>}})
| {{math|''x'', ''y'' ≠ 0}}
|-
| align="center" | {{math|''x''<sup>''p''</sup>}}
| align="center" | <math>\sqrt[p]y</math> (i.e. {{math|''y''<sup>1/''p''</sup>}})
| integer {{math|''p'' > 0}}; {{math|''x'', ''y'' ≥ 0}} if {{math|p}} is even
|-
| align="center" | {{math|''a''<sup>''x''</sup>}}
| align="center" | {{math|[[logarithm|log]]<sub>''a''</sub> ''y''}}
| {{math|''y'' > 0}} and {{math|''a'' > 0}} and {{math|''a'' ≠ 1}}
|-
| align="center" | {{math|''x''[[e (mathematical constant)|''e'']]<sup>''x''</sup>}}
| align="center" | {{math|[[Lambert W function|W]] (''y'')}}
| {{math|''x'' ≥ −1}} and {{math|''y'' ≥ −1/''e''}}
|-
| align="center" | [[trigonometric function]]s
| align="center" | [[inverse trigonometric function]]s
| various restrictions (see table below)
|-
| align="center" | [[hyperbolic function]]s
| align="center" | [[inverse hyperbolic function]]s
| various restrictions
|}

=== Formula for the inverse ===
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse <math>f^{-1} </math> of an invertible function <math>f\colon\R\to\R</math> has an explicit description as

: <math>f^{-1}(y)=(\text{the unique element }x\in \R\text{ such that }f(x)=y)</math>.

This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if {{mvar|f}} is the function

: <math>f(x) = (2x + 8)^3 </math>

then to determine <math>f^{-1}(y) </math> for a real number {{Mvar|y}}, one must find the unique real number {{mvar|x}} such that {{math|1= (2''x'' + 8)<sup>3</sup> = ''y''}}. This equation can be solved:

: <math>\begin{align}
      y         & = (2x+8)^3 \\
  \sqrt[3]{y}   & = 2x + 8   \\
\sqrt[3]{y} - 8 & = 2x       \\
\dfrac{\sqrt[3]{y} - 8}{2} & = x .
\end{align}</math>

Thus the inverse function {{math|''f''<sup> −1</sup>}} is given by the formula

: <math>f^{-1}(y) = \frac{\sqrt[3]{y} - 8} 2.</math>

Sometimes, the inverse of a function cannot be expressed by a [[closed-form formula]]. For example, if {{mvar|f}} is the function

: <math>f(x) = x - \sin x ,</math>

then {{mvar|f}} is a bijection, and therefore possesses an inverse function {{math|''f''<sup> −1</sup>}}. The [[Kepler's equation#Inverse Kepler equation|formula for this inverse]] has an expression as an infinite sum:

: <math> f^{-1}(y) =
\sum_{n=1}^\infty
 \frac{y^{n/3}}{n!} \lim_{ \theta \to 0} \left(
 \frac{\mathrm{d}^{\,n-1}}{\mathrm{d} \theta^{\,n-1}} \left(
 \frac \theta { \sqrt[3]{ \theta - \sin( \theta )} } \right)^n
\right).
</math>