Difference between revisions of "Inverse of a function"

Latest revision as of 13:50, 5 March 2007

The inverse of a function is a function that "undoes" the action of a given function.

For example, consider the function $f$ given by the rule $\displaystyle f(x) = x^3 + 6$ . The function $g(x) = \sqrt[3]{x-6}$ has the property that $f(g(x)) = x$ . In this case, $g$ is called the (right) inverse function of $f$ . Similarly, a function $g$ such that $g(f(x))=x$ is called the left inverse function of $f$ . Typically, the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function. For example, in our example above, $g$ is both a right and left inverse to $f$ on the real numbers.

Often the inverse of a function $f$ is denoted by $f^{-1}$ . Note that the $-1$ does not indicate an exponent.

The inverse of a function $f: A \to B$ with range $C$ is a function $f^{-1}: C \to A$ if and only if $f$ is injective, so that every element in the range is mapped from a distinct element in the domain. When the domain and range are subsets of the real numbers, one way to test this is the horizontal line test: if every horizontal line drawn in the plane intersects the graph of the function in at most one point, the function is injective.

This article is a stub. Help us out by expanding it.

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 The '''inverse of a function''' is a [[function]] that "undoes" the action of a given function.
-For example, consider the function <math>f: \mathbb R \to R</math> given by the rule <math>\displaystyle f(x) = x^2 + 6</math>.  The function <math>g(x) = \sqrt{x-6}</math> has the property that <math>f(g(x)) = x</math>.  In this case, <math>g</math> is called the '''(right) inverse function''' of <math>f</math>.  (Similarly, a function <math>g</math> such that <math>g(f(x))=x</math> is called the '''left inverse function''' of <math>f</math>.  Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.  Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math> (the <math>-1</math> does not indicate a [[exponent]]).
+For example, consider the function <math>f</math> given by the rule <math>\displaystyle f(x) = x^3 + 6</math>.  The function <math>g(x) = \sqrt[3]{x-6}</math> has the property that <math>f(g(x)) = x</math>.  In this case, <math>g</math> is called the '''(right) inverse function''' of <math>f</math>.  Similarly, a function <math>g</math> such that <math>g(f(x))=x</math> is called the '''left inverse function''' of <math>f</math>.  Typically, the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function ''the inverse function''.  For example, in our example above, <math>g</math> is both a right and left inverse to <math>f</math> on the [[real number]]s.
-The inverse of a function is only a function itself iff the original function is [[one-to-one]], or that every element in the [[range]] is paired with a distinct element in the [[domain]]. A way to test this is the [[horizontal line test]], where if a horizontal line can be drawn through the graph of a function and touch two points on the graph, the function is not one-to-one.
+Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>.  Note that the <math>-1</math> does ''not'' indicate an [[exponent]].
+The inverse of a function <math>f: A \to B</math> with [[range]] <math>C</math> is a function <math>f^{-1}: C \to A</math> if and only if <math>f</math> is [[injective]], so that every element in the range is mapped from a distinct element in the [[domain]].  When the domain and range are subsets of the [[real number]]s, one way to test this is the [[horizontal line test]]: if every horizontal line drawn in the plane intersects the graph of the function in at most one point, the function is injective.
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Difference between revisions of "Inverse of a function"

Latest revision as of 13:50, 5 March 2007