22 - LECTURE 22: COMPOSITION AND INVERSES ( 9.5-9.6) There...

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Unformatted text preview: LECTURE 22: COMPOSITION AND INVERSES ( 9.5-9.6) There is no doubt that the first requirement for a composer is to be dead. Arthur Honegger (1892 - 1955) (Ask for students to write down topics they would like covered more deeply.) 1. Identity Function Let A be a non-empty set. There is always at least one bijection from A to A . Namely the function id : A A given by id( a ) = a . This is called the identity function on A , sometimes written id A . 2. Composition Let f : A B and g : B C be functions. (Pictures drawn in class.) We can define a new function h : A C given by the rule h ( a ) = g ( f ( a )). This is called the composition of f and g . We usually write h = g f . Note that we must have the codomain of f match the domain of g . Example 1. Let f : R R be given by f ( x ) = x 2 . Let g : R R be given by g ( x ) = sin( x ). We have ( g f )( x ) = sin( x 2 ). In this case, we can also define ( f g )( x ) = sin 2 ( x ) (since the codomain of g matches the domain of...
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22 - LECTURE 22: COMPOSITION AND INVERSES ( 9.5-9.6) There...

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