functions - Function s The concept of a function is...

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Unformatted text preview: Function s The concept of a function is fundamental in mathematics and is probably best understood in a dynamic way as a process which can be applied to certain input numbers such that each input number gives rise to a specific unique output number. The following "function machine" picture illustrates this idea. The presence of the handle is to suggest that winding it will cause the function machine to produce the required output. A function f has two ingredients: A rule for determining the output value = y ( ) f x for a given input number x , A set of input numbers x called the domain D( f ) of the function f. The range R( f ) of a function f, with a specified domain D( f ), is the set of all values , or output numbers, obtained from inputs taken from the given domain D( f ). R( f ) = { y | = y ( ) f x and x D( f ) }. The vertical line " | " means "such that ", and means " is an element or member of ". The graph of f is the set of points ( ) , x y , where = y ( ) f x and x is a member of D( f ). If the domain of a general function f is not specified, it is taken to be the maximal domain , that is, the largest set of real numbers x for which the function rule = y ( ) f x produces real number values y . For example, let f be the function defined by = ( ) f x 1- x 2 1 . The value of f at a number x is obtained by first squaring the number x , then subtracting 1 and then finally dividing the number obtained into 1. These operations constitute what we could call the " action " of the function f. ( ) f 1 and ( ) f- 1 do not exist because the denominator in 1- x 2 1 is zero when = x 1 or = x- 1. Thus the (maximal) domain of f is: D( f ) = { x | x R and x 1 and x- 1 }. Some examples of evaluation of the function f are as follows. = f 2 3 1- 4 9 1 = 1 - 5 9 = - 9 5 = ( ) f 2 1- ( ) 2 2 1 = 1- 2 1 = 1 = ( ) f- 3 2 1- ( )- 3 2 2 1 = 1- ( )- + 9 6 2 2 1 = 1- 10 6 2 = 1 2 ( )- 5 3 2 = + 5 3 2 2 ( )- 5 3 2 ( ) + 5 3 2 = + 5 3 2 2 ( )- 25 18 = + 5 3 2 14 = ( ) f + a 1 1- ( ) + a 1 2 1 = 1- ( ) + + a 2 2 a 1 1 = 1 + a 2 2 a = f 1 t 1- 1 t 2 1 = 1- 1 t 1 = 1 - 1 t t = t- 1 t = f + 1 1 x 1- + 1 1 x 2 1 = 1 + - 1 1 x 1 = 1 1 x = x = ( ) f ( ) f x f 1- x 2 1 = 1- 1- x 2 1 2 1 = 1- 1 ( )- x 2 1 2 1 = 1 - 1 ( )- x 2 1 2 ( )- x 2 1 2 = ( )- x 2 1 2- 1 ( )- x 2 1 2 = ( )- x 2 1 2- 1 ( )- + x 4 2 x 2 1 = ( )- x 2 1 2- 2 x 2 x 4 = ( )- x 2 1 2 x 2 ( )- 2 x 2 Example 1 : Suppose that the function f is described by = ( ) f x x 2 , with the domain specified to be: D( f ) = { x | x R and - 2 x 2 }, that is, the domain D( f ) is the closed interval [ ] ,- 2 2 ....
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functions - Function s The concept of a function is...

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