Inverse function: Difference between revisions

Revision as of 03:54, 28 April 2022

An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse.

Definition

A function $g$ is the inverse function of $f$ if $f(g(x))=x$ for each value of $x$ in the domain of $g$ , and $g(f(x))=x$ for each value of $x$ in the domain of $f$ . The function $g$ is denoted as $f$ ^-1 ("inverse of f").

Notation

Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, $f<sup>-1</sup>(x)$ is not equal to 1/f(x).

Relevant observations

If g is the inverse function of f, then f is the inverse function of g.
The domain of f^-1 is the range of f and the range of f^-1 is the domain of f.
A function may not have an inverse function, but if it has, the inverse function is unique.

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 == Definition ==
-A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as f<sup>-1</sup> ("inverse of f").
+A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as <math>f</math><sup>-1</sup> ("inverse of f").
 === Notation ===
-Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f</math><sup>-1</sup>(x) is not equal to 1/f(x).
+Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f<sup>-1</sup>(x)</math> is not equal to 1/f(x).
 == Relevant observations ==