2.1 - MATH 1081  Wednesday, January 19   ...

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Unformatted text preview: MATH 1081  Wednesday, January 19    Section 2.1    PROPERTIES OF FUNCTIONS    Homework #1 (due 1/24):   Section 1.1 #36, 64, 70, 72                  Section 2.1 #36, 44, 46, 64    Starting today, clicker questions will be asked in each lecture  session.  Remember that it is ½ point for an incorrect answer  and 1 point for a correct answer,  also ½ point for check‐in and  ½ point for check‐out.    To ready your clicker, press and hold the On/Off button until  the Power light flashes.  Then enter the 2‐key frequency code  (one letter at a time) as indicated on the sign posted at the  front of the room.  At that point you should see the Vote Status  light flash green.    Clicker Check­in:  Press “A” now to check in.  The Vote Status light should again flash green to indicate that  your vote has been received. As we discussed briefly during the last lecture, for much of this  class, we will be interested in determining the rate of change of  a graph at a certain point.  To do this formally, we use the  language of functions.    Definition: Function  A function is a rule that assigns to each element from one  set exactly one element from another set.      For example, the rule that pairs each student in the class with  their birthday is a function, since every student has one and  only one birthday.  However, the rule that pairs each day of the  year with the student that was born on that day may not be.   Why?  The notation that we typically use when working with a  function looks like         f ( x ) ,  read “f of x”,    where  f  is the name of the function,  x  is an element from one  set (input), and the value of  f ( x )  is the corresponding element  from the other set (output).    For example, the equation  f ( 3) = 7  indicates that for the  function rule named  f , the number 3 is assigned to the number  7.    Keep in mind, it is still okay for other “input” values to also be  paired with 7, so  f ( −5 ) = 7  can also be true.  In most cases, the rule that is to be applied is written as a  mathematical expression.  For example,          f ( x) = x2 + 2x − 8    indicates that for the function named  f  the rule will be to take  the input number  x  square it, add to that 2 times it, and then  subtract from that total 8 to get the corresponding output.    So, when  x = 3, we get  32 + 2 ( 3) − 8 = 9 + 6 − 8 = 7 .  This implies  that  f ( 3) = 7 .    2 Similarly,  f ( −5 ) = ( −5 ) + 2 ( −5 ) − 8 = 25 − 10 − 8 = 7 .  Example:  For each function, find   (a)  f ( 4 ) ,     (b)  f ( −1) ,     1.   f ( x ) = 2 x 2 − 5 x + 1        x +1 2.   f ( x ) =   x−2       3.   f ( x ) = 3x −1     (c)  f ( 2 a ) .  Definition:  Domain  The set of all “input” values of a function is called its  domain.      For a general function, unless otherwise stated, we will  assume that the domain is the largest set of real numbers  for which the expression defining the function is defined.    When finding the domain of a function, we must avoid  1) division by zero  2) square roots (or any even index) of a negative number  3) logarithms of a non‐positive number.    Example:  Determine the domain of the function.          1.   f ( x ) = 3x + 7         x+3       2.   f ( x ) = 2   x −4       x−2       3.   f ( x ) = 2   x − 4x − 5 Definition:  Range  The set of all “output” values of a function is called its  range.      Generally, it is difficult to determine the range of a function just  from the expression defining it.  However, if we have a graph of  that function, it is quite easy to determine by examining which  y ‐values are attained.    To consider the graph of a function, we need to identify the  value of the function  f ( x )  with the  y ‐coordinate on the graph.   That is,  y = f ( x ) .    For example, if we consider the function  f ( x ) = x 2 + 2 x − 8 , and  make a table of values, and plot those points, what shape  results?    y = f ( x )  x  0  ‐8  1  ‐5  ‐1  ‐9  2  0  ‐2  ‐8  What is the domain and range of the function shown here?      As a final note, as the definition of a function implies, when  graphing  y = f ( x ) , for each  x  in the domain of  f , there will be  exactly ONE corresponding value of  y .    So, if we draw a vertical line through that value of  x , it will  intersect the graph of  f  exactly one time.  If when we draw a  vertical line through ANY value of  x  and it intersects the graph  more than once, this cannot be the graph of a function of  x .    What does it mean if a vertical line drawn through a value of  x   doesn’t intersect the graph at all?  Clicker Check­Out:  Press “E” to check out now.      Tomorrow in recitation:  Prerequisite Skills Exam      NEXT TIME – Section 3.1  ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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