ࡱ>  STUAy AbjbjVV 7B<<8PNN   8Et  QMl '4'''N0Z00PPPPPPP$SVP2J0N022P'']P;;;2t''P;2P;;6LN'  E4MPP0 QM W98z W8N WN0X1^;1L2000PP: 000 Q2222 W000000000N W:  Section II: Functions, Inequalities, and the Absolute Value  Module 1: Introduction to Functions A function is a special type of binary relation. So before we discuss what a function is, we need to define binary relation. DEFINITION: A binary relation is a rule that establishes a relationship between two sets. (Note that this is an informal definition.)  The two sets involved in a binary relation play different roles; these roles are determined by the rule of the relation (see the first example). The two roles that the sets play are the first set or the set of inputs and the second set or the set of outputs. The set of inputs is called the domain; the set of outputs is called the range.  EXAMPLE: a. If the rule of the relation is Associate each person with his/her social security number then the domain is the set of people and the range is the set of social security numbers. b. If the rule of the relation is Associate each social security number with the person who has been assigned to that number, then the domain is the set of social security numbers and the range is the set of people.   EXAMPLE: Consider a binary relation defined on the following two sets: Set of inputs (domain): {Peter, Kandace, Elijah, Logan} Set of outputs (range): {Cat, Dog, Mouse} The rule that relates these sets is that each person (in the set of inputs) is related to an animal (from the set of outputs) that he or she has as a pet. The arrow diagram below defines this relation:  EMBED PBrush  The arrow diagram tells us, for example that Logan is related to Dog (since there is an arrow from Logan to Dog), and this means that Logan has a dog. We also see that Logan is related to Mouse, so Logan also has a mouse. Another way to convey the information contained in a relation is to use ordered pairs. To represent the fact that Logan is related to Dog we use the ordered pair  EMBED Equation.DSMT4 . The first term in each ordered pair is an element of the set of inputs and the second term in each ordered pair is an element of the set of outputs. The entire relation represented by the arrow diagram above can be represented by the set of ordered pairs below:  EMBED Equation.DSMT4  DEFINITION: A function is a binary relation in which each element in the domain corresponds to exactly one element in the range.   EXAMPLE: The arrow diagram below represents a binary relation on the sets {Steve, Nancy, Kyra, Atticus} and {Boy, Girl}. The rule that defines this relation is Associate each person with his/her gender. This relation is a function since each person has exactly one gender. On the arrow diagram we can tell that this is a function since there is exactly one arrow coming out of each element in the domain. (Notice that in the example above, more than one arrow comes from some elements in the domain, so the diagram does not represent a function.)  EMBED PBrush  We can represent this function by the following set of ordered pairs:  EMBED Equation.DSMT4   EXAMPLE: Determine whether each of the following relations describes a function with domain  EMBED Equation.3  and range  EMBED Equation.3 . a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  SOLUTIONS: a.  EMBED Equation.DSMT4  represents a function because each element in the domain corresponds to exactly one element in the range. b.  EMBED Equation.DSMT4  does not represent a function because the number 1 corresponds to two different elements of the range. c.  EMBED Equation.DSMT4  is a function because each element in the domain corresponds to exactly one element in the range. (Note that it is okay to repeat elements in the range.)   EXAMPLE: The table below defines the function f, where the inputs come from  EMBED Equation.3  and the outputs come from  EMBED Equation.3 . Thus, the set A is the domain of f while the set B is the range of f, which we can represent with the notation  EMBED Equation.3 . domain of f(5(4(3(2(1012345range of f4(3(541309(3213 If someone were to ask you, Where does f send (2? you would (hopefully) say, f sends (2 to 4 (since there is a column in the table showing that the element (2 in the domain is related to 4 in the range). Function notation offers another way to communicate this information:  EMBED Equation.3  means f sends (2 to 4. Use the table above to evaluate the following: a. Evaluate  EMBED Equation.3 .b. Evaluate  EMBED Equation.3 .c. Evaluate  EMBED Equation.3 .HYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_a.swf"CLICK HERE FOR ANSWER HYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_b.swf" CLICK HERE FOR ANSWERHYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_c.swf"CLICK HERE FOR ANSWER Sometimes we dont know what the input is. In such a case, we can call the input x (or any other variable), where x represents a generic element of the domain; then  EMBED Equation.3  represents the element in the range that x is associated with under the function f. In this example, the domain of f is the set A and the range is the set B. Thus,  EMBED Equation.3  and  EMBED Equation.3  With this new notation, we can rename the rows in our table: x(5(4(3(2(1012345 EMBED Equation.3 4(3(541309(3213 Solve the following for x. d. Solve  EMBED Equation.3 .e. Solve  EMBED Equation.3 .f. Solve  EMBED Equation.3 .HYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_d.swf"CLICK HERE FOR ANSWERHYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_e.swf"CLICK HERE FOR ANSWERHYPERLINK "http://spot.pcc.edu/~phaberma/Learning_Objects/functionalNotation/sec2mod2_f.swf"CLICK HERE FOR ANSWER  EQUATIONS AS FUNCTIONS Functions are often defined by equations. For example, the equation  EMBED Equation.3  defines a function, where the inputs are represented by x and the outputs are represented by y. It is often helpful to write equations like  EMBED Equation.3  in function notation:  EMBED Equation.3  Similarly, the equation  EMBED Equation.3  can be written in function notation as  EMBED Equation.3 . This is read: g of t equals 3t minus 5. Caution: The notation  EMBED Equation.3  does not mean multiplication of f by x. Function notation involving  EMBED Equation.3  is more compact than notation involving  EMBED Equation.3 . For example: With function notation: If  EMBED Equation.3 , then EMBED Equation.3 . Without function notation: If  EMBED Equation.3 , then the value of y is 8 when x is 5. The independent variable refers to the variable representing possible values in the domain, and the dependent variable refers to the variable representing possible values in the range. Thus in our usual ordered pair notation EMBED Equation.3 , x is the independent variable and y is the dependent variable. It is customary to plot the independent variable on the horizontal axis and the dependent variable on the vertical axis.  EXAMPLE: Find the indicated function value.  EMBED Equation.3  if  EMBED Equation.3   EMBED Equation.3  if  EMBED Equation.3   EMBED Equation.3  if  EMBED Equation.3   EMBED Equation.3  if  EMBED Equation.3  SOLUTIONS:  EMBED Equation.3   EMBED Equation.DSMT4   EMBED Equation.3   EMBED Equation.3   HYPERLINK "http://pcc.mediasite.com/pcc/Viewer/Viewers/Viewer240TL.aspx?mode=Default&peid=a4040b9c-c513-4d4e-9601-0561aa8b6733&pid=8effb0e5-4961-476e-8bde-6ecb577ffd33&playerType=WM7" INCLUDEPICTURE "http://tbn0.google.com/images?q=tbn:__P5nwIFMBs1cM:http://www.boostmobilecommunity.com/CMSimages/community/blog/movie-reel-2.jpg" \* MERGEFORMATINET HYPERLINK "http://mediasite.pcc.edu/Mediasite/Viewer/?peid=a4040b9cc5134d4e96010561aa8b67331d"CLICK HERE FOR A FUNCTION NOTATION EXAMPLE  HYPERLINK "http://pcc.mediasite.com/pcc/Viewer/Viewers/Viewer240TL.aspx?mode=Default&peid=eabc8d9b-0648-4afc-9d24-a2839cef035d&pid=8f3818c1-2ce6-4d9b-b17e-7d20165261e9&playerType=WM7" INCLUDEPICTURE "http://tbn0.google.com/images?q=tbn:__P5nwIFMBs1cM:http://www.boostmobilecommunity.com/CMSimages/community/blog/movie-reel-2.jpg" \* MERGEFORMATINET HYPERLINK "http://mediasite.pcc.edu/Mediasite/Viewer/?peid=a3cc164f8f434f0881c1577203a112cd1d"CLICK HERE FOR ANOTHER FUNCTION NOTATION EXAMPLE  EXAMPLE: Suppose  EMBED Equation.DSMT4 . Find  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . SOLUTIONS:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4    EXAMPLE: Suppose  EMBED Equation.DSMT4 . Find  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . SOLUTIONS:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4    EXAMPLE: The function f is graphed in Figure 1 below. (Note that there is an arrow on the left-hand side of the graph since the function doesnt end when the graph ends. If the graph ends, it will be denoted with a filled-in circle, as there is on the right-hand side.) Determine the following: a.  EMBED Equation.3  b. the domain of f c. any x-value(s) for which  EMBED Equation.3  d. any x-value(s) for which  EMBED Equation.3  e. the range of f  Figure 1: Graph of  EMBED Equation.3 . SOLUTIONS: a.  EMBED Equation.3  which can be determined because the point  EMBED Equation.3  is on the graph. b. The domain of f is  EMBED Equation.3  since the graph suggests that there is a point above or below every x-value less than or equal to 3; every x-value less than or equal to 3 is "used" by the function so it's in the domain. c.  EMBED Equation.3  when  EMBED Equation.3  because  EMBED Equation.3 . (Notice that we use  EMBED Equation.DSMT4  to denote the fact that we are approximating the answer.) d.  EMBED Equation.3  when  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . e. The range of f is  EMBED Equation.3  since the graph suggests that there is a point to the left or right of every y-value greater than or equal to  EMBED Equation.DSMT4 ; every y-value greater than or equal to  EMBED Equation.DSMT4  is "used" by the function so it's in the range.   EXAMPLE: The function p is graphed in Figure 2 below. Find the following: a.  EMBED Equation.3 . b. the domain of p. c. all x-values for which  EMBED Equation.3 . d. the range of p.  Figure 2: Graph of  EMBED Equation.3 . SOLUTIONS: a.  EMBED Equation.3  because the point  EMBED Equation.3  is on the graph. Note that there is a hole at the point  EMBED Equation.3 , so the point  EMBED Equation.3  is NOT on the graph. b. The domain of p is  EMBED Equation.3  (in interval notation) or  EMBED Equation.DSMT4  (in set-builder notation). c. The x-values for which  EMBED Equation.3  are  EMBED Equation.3  and  EMBED Equation.3 . d. The range of p is  EMBED Equation.3  (in interval notation). Recall that the definition of a function requires that each input has only one corresponding output. Since the inputs are x-values and the outputs are y-values, a function can have only one y-value for each x-value. If we translate this information to the context of the graph of a function, we quickly realize that (since each x-value is represented by a vertical line) the graph of a function can have at most one point on each vertical line. (If a vertical line  EMBED Equation.DSMT4  passes through a graph more than one time, then the x-value a has more than one corresponding output, so the graph cannot represent a function.) This leads us to the Vertical Line Test: The Vertical-Line Test If it is possible for a vertical line to cross a graph more than once, then the graph does not represent a function.   example: In which of the following graphs is y a function of x ?  Figure 3 Figure 4 Figure 5 SOLUTION: The graph in Figure 4 is the only one that passes the vertical line test so this is the only graph in which y a function of x.   example: The function  EMBED Equation.3  is defined in Figure 6, the function  EMBED Equation.3  is defined in Table 1, and  EMBED Equation.3 .  Figure 6: Graph of  EMBED Equation.3  Table 1x EMBED Equation.3 2315021127 Find each of the following: a.  EMBED Equation.DSMT4  e.  EMBED Equation.3  b.  EMBED Equation.3  f. x if  EMBED Equation.3  c.  EMBED Equation.3  g. x if  EMBED Equation.3  d.  EMBED Equation.DSMT4  h. x if  EMBED Equation.3  SOLUTIONS:  EMBED Equation.3  (using the table)  EMBED Equation.3  (using the graph) Note that  EMBED Equation.3  since  EMBED Equation.3  is not a point on the graph. The solid circles represent points on the graph, whereas the open circle at  EMBED Equation.3  represents a hole in the graph.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.DSMT4  Since  EMBED Equation.3 , if  EMBED Equation.3 , then  EMBED Equation.3 . Look for an output value of  EMBED Equation.3  in the table.  EMBED Equation.DSMT4  To solve this equation we need to find an output value of 0 on the graph of  EMBED Equation.3 . (So we are looking for the x-intercepts on the graph of  EMBED Equation.3 ). Since  EMBED Equation.3  and  EMBED Equation.3 , if  EMBED Equation.3 , then  EMBED Equation.3  or  EMBED Equation.3 . Another way of writing this using set notation is  EMBED Equation.3 , or we could say, The solution set for the equation  EMBED Equation.3  is  EMBED Equation.3 .  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