Difference between revisions of "Inverse of a function"
(This needs a lot of work. I mostly ripped it off the introduciton section of function, and it could use work there, too.) |
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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 | + | 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. |
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+ | 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]]. | ||
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+ | 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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Latest revision as of 12:50, 5 March 2007
The inverse of a function is a function that "undoes" the action of a given function.
For example, consider the function 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. For example, in our example above,
is both a right and left inverse to
on the real numbers.
Often the inverse of a function is denoted by
. Note that the
does not indicate an exponent.
The inverse of a function with range
is a function
if and only if
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.
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