Editing Inverse function (section)

=== 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>