inverse_functions

inverse_functions - Inverse Functions One-to-one functions...

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Inverse Functions One-to-one functions Let f be a function with domain and range which are subsets of the set of real numbers. Given an element y in the range of f, there must be an element x in the domain of f, such that = ( ) f x y . For a given element y there may be, in general, more than one element x such that = ( ) f x y . For example, for the function = ( ) f x x 2 with domain the set of all real numbers, the number 4 is in the range of f because = ( ) f 2 4. However, we also have = ( ) f - 2 4, that is, there are two numbers = x 2 and = x - 2 such that = ( ) f x 4. For the function = ( ) f x x 3 , given any element y in the range of f, there is exactly one element x in the domain of f such that = ( ) f x y . This function f is an example of a one-to-one function. A function f if is called one-to-one if, for every element y in the range of f, there is exactly one element x in the domain of f such that = ( ) f x y . A function f is one-to-one when every line parallel to the x axis meets the graph of f at no more than one point . Note that any linear function f where = ( ) f x + a x b and a 0 is a one-to-one function.
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The inverse function of a one-to-one function The square root function g where = ( ) g x x can only be understood by relating it to the squaring function f where = ( ) f x x 2 . When we look for the square root of a number, say 169, we try to find a number which is greater than or equal to zero, and with the property that, if we square it, the result is 169. We look for a number x , where x > 0 and such that = ( ) f x 169. Since = 13 2 169, we have = 169 13. The square root function provides an inverse function for the squaring function, as long as we only consider numbers which are greater than or equal to zero. Given a one-to-one function f, with domain A and range B , the inverse function of f, denoted by f ( ) - 1 , is the function with domain B and range A , which reverses the action of f, that is, f ( ) - 1 = ( ) y x exactly when = y ( ) f x ------- (i). Interchanging x and y in (i) gives: = y f ( ) - 1 ( ) x exactly when = x ( ) f y . _______________ In terms of the composition of functions, we have: ( f ( ) - 1 o f)( x ) = f ( ) - 1 = ( ) ( ) f x x for any number x in the domain A of f ( which is the range of f ( ) - 1 ), ( f o f ( ) - 1 )( y ) = f ( f ( ) - 1 ( y )) = y for any number y in range B of f ( which is the domain of f ( ) - 1 ). Another way to phrase the first preceding point is to say that the composition f ( ) - 1 o f is the identity function I A on the set A , that is, I A is the function with domain and range the set A with defining formula = ( ) I A x x , for all members of the domain A . Similarly, the second point amounts to the fact that the composition f o f ( ) - 1 is the identity function I B on the set B . On any scientific calculator, the inverse of a particular function is usually obtained by using the shift key, or second function key. Thus tan ( ) - 1 is an inverse for the tangent function (at least when the tangent function is restricted to to the interval of numbers from - π 2 to π 2 , so that it is a one-to-one function) .
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