Inverse Function
Given a function f(x), its inverse f^(-1)(x) is defined by
| f(f^(-1)(x))=f^(-1)(f(x))=x. |
(1)
|
Therefore, f(x) and f^(-1)(x) are reflections about the line y=x. In the Wolfram Language, inverse functions are represented using InverseFunction [f].
As noted by Feynman (1997), the notation f^(-1)x is unfortunate because it conflicts with the common interpretation of a superscripted quantity as indicating a power, i.e., f^(-1)x=(1/f)x=x/f. It is therefore important to keep in mind that the symbols sin^(-1)z, cos^(-1)z, etc., refer to the inverse sine, inverse cosine, etc., and not to 1/sinz=cscz, 1/cosz=secz, etc.
A function f admits an inverse function f^(-1) (i.e., "f is invertible") iff it is bijective. However, inverse functions are commonly defined for elementary functions that are multivalued in the complex plane. In such cases, the inverse relation holds on some subset of the complex plane but, over the whole plane, either or both parts of the identity f^(-1)(f(z))=f(f^(-1)(z))=z may fail to hold. A few examples are illustrated above and in the following table. In the table 0<b<a and c in Z.
An additional counterintuitive property of inverse functions is that
![画像: sqrt(z)sqrt(1/z)={-1 for I[z]=0 and R[z]<0; undefined for z=0; 1 otherwise,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/InverseFunction/NumberedEquation2.svg){kind=link}
so the expected identity does not hold along the negative real axis.
See also
Bijective, Composition, Inverse, Inverse Function Theorem, Inverse Hyperbolic Functions, Inverse Trigonometric Functions, Series Reversion Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
References
Feynman, R. P. "He Fixes Radios by Thinking!" In 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, p. 12, 1997.Jeffreys, H. and Jeffreys, B. S. "Inverse Functions." §1.066 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 22-23, 1988.Ritt, J. F. "Elementary Functions and Their Inverses." Trans. Amer. Math. Soc. 27, 68-90, 1925.Referenced on Wolfram|Alpha
Inverse FunctionCite this as:
Weisstein, Eric W. "Inverse Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseFunction.html