Difference between revisions of "Inverse of a function"
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The '''inverse of a function''' is a [[function]] that "undoes" the action of a given function. | 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>. | + | 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]]). |
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| + | 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. | ||
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Revision as of 22:28, 4 March 2007
The inverse of a function is a function that "undoes" the action of a given function.
For example, consider the function 
given by the rule 
. The function 
has the property that 
. In this case, 
is called the (right) inverse function of 
. (Similarly, a function such that 
is called the left inverse function of . 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 is denoted by 
(the 
does not indicate a exponent).
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.
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