Piecewise Functions - ClassZone

1 Page 1 of 2114 Chapter 2 Linear Equations and FunctionsPiecewise FunctionsREPRESENTINGPIECEWISEFUNCTIONSU p to now in this chapter a function has been represented by a single equation. Inmany real-life problems, however, Functions are represented by a combination ofequations, each corresponding to a part of the domain. Such Functions are calledFor example, the Piecewise function given by (x) = is defined by two equations. One equation gives the values of (x) when xis less thanor equal to 1, and the other equation gives the values of (x) when xis greater than a Piecewise FunctionEvaluate (x) when (a) x= 0, (b) x= 2, and (c) x= 4. (x) = SOLUTIONa. (x) = x+ 2 Because 0 <2, use first equation. (0) = 0+ 2 = 2 Substitute 0 for (x) = 2x+ 1 Because 2 2, use second equation. (2) = 2(2) + 1 = 5 Substitute 2 for (x) = 2x+ 1 Because 4 2, use second equation. (4) = 2(4) + 1 = 9 Substitute 4 for a Piecewise FunctionGraph this function: (x) =SOLUTIONTo the left of x= 1, the graph is given by y= 12 x+ 23.

2 To the right of and including x= 1, the graph is given by y= x+ graph is composed of two rays with common initialpoint (1, 2). 12 x+ 32 , if x<1 x+ 3, if x 1 EXAMPLE 2x+ 2, if x< 22x+ 1, if x 2 EXAMPLE 12x 1, if x 13x+ 1, if x>1piecewise piecewisefunctions to model real-lifequantities, such as theamount you earn at a summerjob in Example 6. To solve real-lifeproblems, such as determin-ing the cost of ordering silk-screen T-shirts in Exs. 54 and you should learn itGOAL2 GOAL1 What you should (1, 2)Page 1 of Functions115 graphing a Step FunctionGraph this function: (x) = SOLUTIONThe graph of the function is composed of four line segments. For instance, the firstline segment is given by the equation y= 1 and represents the graph when xisgreater than or equal to 0 and less than .. The function in Example 3 is called a because its graph resembles a set of stair steps. Anotherexample of a step function is the greatest integer function is denoted by g(x) = x.

3 For every realnumber x, g(x) is the greatest integer less than or equalto x. The graph of g(x) is shown at the right. Note that inExample 3 the function could have been written as (x) = x + 1, 0 x< a Piecewise FunctionWrite equations for the Piecewise function whose graph is the left of x= 0, the graph is part of the line passingthrough ( 2, 0) and (0, 2). An equation of this line isgiven by:y= x+ 2To the right of and including x= 0, the graph is part of the line passing through (0, 0)and (2, 2). An equation of this line is given by:y= x The equations for the Piecewise function are: (x) = Note that (x) = x+ 2 does not correspond to x= 0 because there is an opendot at(0, 2), but (x) = xdoescorrespond to x= 0 because there is a soliddot at (0, 0).x+ 2, if x<0x,if x 0 EXAMPLE 4step function11yxThe solid dot at (1, 2)indicates that (1) open dot at (1, 1)indicates that (1) , if 0 x< 12, if 1 x< 23, if 2 x< 34, if 3 x< 4 EXAMPLE 321yx31yx(0, 2)( 2, 0)(0, 0)(2, 2)Page 1 of 2116 Chapter 2 Linear Equations and FunctionsUSINGPIECEWISEFUNCTIONS INREALLIFEU sing a Step and graph a Piecewise function for theparking charges shown on the are the domain and range of the function?

4 Times up to one half hour, the chargeis $3. For each additional half hour (orportion of a half hour), the charge is anadditional $3 until you reach $8. Let trepresent the number of hours you Piecewise function and graph domain is 0 < t 12, and the range consists of 3, 6, 8 Using a Piecewise FunctionYou have a summer job that pays time and a half for overtime. That is, if you workmore than 40 hours per week, your hourly wage for the extra hours is times yournormal hourly wage of $ and graph a Piecewise function that gives your weekly pay Pin terms ofthe number h of hours you much will you get paid if you work 45 hours? up to 40 hours your pay is given by 7h. For over 40 hours your pay is given by:7(40) + (7)(h 40) = 140 The Piecewise function is:P(h) = The graph of the function is shown. Note that for upto 40 hours the rate of change is $7 per hour, but forover 40 hours the rate of change is $ per find how much you will get paid for working 45 hours, use the equationP(h) = (45) = (45) 140 = You will earn $ ,if 0 h 140, if h>40 EXAMPLE 6 EXAMPLE 5 GOAL2 REALLIFEREALLIFEU rban ParkingREALLIFEREALLIFEW agesPay (dollars)0P(h)hHours20030010040200(40, 280)Summer Job21t1211 Time (hours)Cost (dollars)84 (t)0 Weeknight Rates (t) =3, if 0 < t , if < t 18, if 1 < t 123 Garage Rates (Weekends)$3 per half hour$8 maximum for 12 hoursPage 1 of Piecewise function and step function.

5 Give an example of back at Example 3. What does a solid dot on the graph of a step functionindicate? What does an open dot indicate?Tell whether the statement is Trueor False. the graph of a Piecewise function, the separate pieces are always (x) = can be rewritten as (x) = 2 x , 1 x< (x) = for the given value of 13 2 Graph the (x) = 10. (x) = equations for the Piecewise function whose graph is weekday parking rates for a garage are shown. Write and graph apiecewise function for the weekday parkingcharges at that the function for the given value of ( 4)14. ( 2)15. (0)16. (5) (1) ( 10) (6) (0)GRAPHINGFUNCTIONSG raph the (x) = 22. (x) = 23. (x) = 24. (x) = 25. (x) = 26. (x) = x 8, if x<9 13 x 2, if x 93x 14, if x 4 2x+ 6, if x>4 x,if x>2x 4, if x 22x+ 13, if x 5x+ 12 , if x< 5x+ 6,if x 3 23 x 3, if x> 32x,if x 1 x+ 3, if x<1 PRACTICEANDAPPLICATIONS4, if 0 x< 25, if 2 x< 46, if 4 x< 62x+ 1, if x<1 x+ 4, if x 13x 1, if x 42x+ 7, if x> 42, if 1 x< 24, if 2 x< 36, if 3 x< 4 GUIDEDPRACTICEx22y(0, 6)(3, 2)(8, 0)Vocabulary Check Concept Check Skill Check STUDENTHELPE xtra Practice to help you masterskills is on p.

6 942. (x) = h(x) = 12 x 10, if x 6 x 1, if x>65x 1, if x< 2x 9, if x 2 Garage Rates (Weekdays)$3 per half hour$18 maximum for 12 hoursSTUDENTHELPHOMEWORK HELPE xample 1:Exs. 13 20 Example 2:Exs. 21 26 Example 3:Exs. 27 32 Example 4:Exs. 35 40 Examples 5 and 6:Exs. 50 59Ex. 11 Page 1 of 2118 Chapter 2 Linear Equations and FunctionsGRAPHINGSTEPFUNCTIONSG raph the step (x) = 28. (x) = 29. (x) = 30. (x) = SPECIALSTEPFUNCTIONSG raph the special step function. Then explainhow you think the function got its FUNCTION (x)= x= (x) = ROUND(x) = back at Example 2. How would the graph of thefunction change if < was replaced with and was replaced with >? Explainyour back at Example 3. How would the graph of thefunction change if each was replaced with < and each < was replacedwith ? Explain your equations for the Piecewise functionwhose graph is many graphing calculators x isdenoted by int(x). Use a graphing calculator to graph the (x) = x (x) = 2x (x) = x (x) = x+ 3 (x) = 6 x (x) = 3x + (x) = 4 x+ 7 (x) = x (x) = 3 x 2 + 5yx 1211xyx22yx11y1yx1yx11.

7 1, if x< , if x< , if x< ..1, if 0 < x 12, if 1 < x 23, if 2 < x 3..4, if 10 < x 86, if 8 < x 68, if 6 < x , if 4 < x 210, if 2 < x 0 1, if 0 x< 1 3, if 1 x< 2 5, if 2 x< 3 7, if 3 x< 4 9, if 4 x< , if 4 x< , if 2 x< , if 1 x< 3 , if 3 x< 63, if 1 x< 25, if 2 x< 48, if 4 x< 910, if 9 x< 12 KEYSTROKE HELPV isit our Web see keystrokes forseveral models 1 of Functions119 POSTALRATESIn Exercises 50 and 51, use the following of January 10, 1999, the cost C(in dollars) of sending next-day mail using theUnited States Postal Service, depending on the weight x(in ounces) of a package upto five pounds, is given by the function (x) = the the domain and range of the Exercises 52 and 53, use the function given for thecost C(in dollars) of making xphotocopies at a copy (x)= the your graph to explain why it would not be cost-effective to make 450 Exercises 54 and 55, use the following silk-screen shop charges.

8 An initial charge of $20 to create the silk screen $ per shirt for orders of 50 or fewer shirts $ per shirt for orders of more than 50 a Piecewise function that gives the cost Cfor an order of x the Exercises 56 and 57, use the following amount of Social Security tax you pay, part of your Federal InsuranceContributions Act (FICA) deductions, depends on your annual income. As of 1999you pay of your income if it is less than $72,600. If your income is at least$72,600, you pay a fixed amount of $ and graph a Piecewise function that gives the Social Security much Social Security tax do you pay if you make $30,000 per year?SNOWSTORMIn Exercises 58 and 59, use the following a nine hour snowstorm it snows at a rate of 1 inch per hour for the first twohours, at a of rate of 2 inches per hour for the next six hours, and at a rate of 1 inchper hour for the final and graph a Piecewise function that gives the depth of the snow duringthe many inches of snow accumulated from the storm?

9 , if 0 < x , if 26 x , if 101 x , if 501 , if 0 < x , if 8 < x , if 32 < x , if 48 < x , if 64 < x 80 DATA UPDATE of United States Postal Service data at INTERNETSNOWSTORM By weighing snow atthe end of a snowstorm youcan determine the watercontent of the snow. Thisinformation is one of thefactors used to determineavalanche ONAPPLICATIONSHOMEWORK HELPV isit our Web help with problemsolving in Exs. 52 and UPDATE of Social Security Administration data at INTERNETPage 1 of 2120 Chapter 2 Linear Equations and FunctionsQUANTITATIVECOMPARISONIn Exercises 60 and 61, choose the statementthat is true about the given quantities. AThe quantity in column A is greater. BThe quantity in column B is greater. CThe two quantities are equal. DThe relationship cannot be determined from the given time t(in minutes) that a person may safely scuba divewithout having to decompress while surfacing is determined by the depth d(infeet) of the dive.

10 Using the information below, write and graph a piecewiseinequality that describes the time limits for scuba divers at various depths. For depths from 40 feet (the minimum depth requiring decompression) to 53 13 feet, the time must not exceed 600 minutes minus ten times the depth. For depths greater than 53 13 feet to less than 90 feet, the time must not exceed120 minutes minus the depth. For depths from 90 feet to 130 feet (the maximum safe depth for a recreationaldiver), the time must not exceed 75 minutes minus one half the the equation. (Review for )63.|9 + 4x| = 1564.|7x+ 3| = 1165.|21 2x| = 966.|2x+ 8| = 167.| 12 x 5|= 1168.|1 34 x|= 6 SCATTERPLOTSDraw a scatter plot of the data. Then tell whether the datahave a positive, a negative, or relatively no correlation. (Review ) be comfortable, sleeping bags rated for 40 F inches of insulation, and those rated for 40 F have inches. Write a linearmodel for the amount aof insulation needed to be comfortable at temperature much insulation would you need to be comfortable at 0 F?