7 Rtionl Functions 7. Inverse Vrition 7. Grphing Rtionl Functions 7.3 Multipling nd Dividing Rtionl Epressions 7. Adding nd Subtrcting Rtionl Epressions 7. Solving Rtionl Equtions Cost of Fuel (p. 397) Glpgos Penguin (p. 38) Lightning Strike (p. 37) SEE the Big Ide 3-D Printer (p. 39) Volunteer Project (p. 3)
Mintining Mthemticl Proficienc Adding nd Subtrcting Rtionl Numbers Emple Find the sum 3 + 3. 3 + 3 = 9 + = 9 + = Emple Find the difference 7 8 ( 7 8 ( 8) = 7 8 + 8 = 7 + 8 = 8 = 3, or 8). Rewrite using the LCD (lest common denomintor). Write the sum of the numertors over the common denomintor. Add. Add the opposite of 8. Write the sum of the numertors over the common denomintor. Add. Simplif. Evlute.. 3 + 3. ( ).. 7 + 7 + 7 7 3. 7 9 9 3. 0 3 + Simplifing Comple Frctions. Emple 3 Simplif Simplif. 7. 3 8 = = = = 8 8. Rewrite the quotient. Multipl b the reciprocl of. Multipl the numertors nd denomintors. Simplif. 7 9. 3 3 + 0. ABSTRACT REASONING For wht vlue of is the epression undefined? Eplin our resoning. Dnmic Solutions vilble t BigIdesMth.com 37
Mthemticl Prctices Specifing Units of Mesure Core Concept Converting Units of Mesure Mthemticll profi cient students re creful bout specifing units of mesure nd clrifing the reltionship between quntities in problem. To convert from one unit of mesure to nother unit of mesure, ou cn begin b writing the new units. Then multipl the old units b the pproprite conversion fctors. For emple, ou cn convert 0 miles per hour to feet per second s follows. old units 0 mi h =? ft sec 0 mi h h 0 min min 80 ft 80 ft = 0 sec mi 0 sec 88 ft = sec new units Monitoring Progress Converting Units of Mesure You re given two job offers. Which hs the greter nnul income? $,000 per er $ per hour One w to nswer this question is to convert $ per hour to dollrs per er nd then compre the two nnul slries. Assume there re 0 hours in work week. dollrs h =? dollrs r dollrs 0 h weeks,70 dollrs h = week r r The second offer hs the greter nnul slr. Write new units. Multipl b conversion fctors.. You drive cr t speed of 0 miles per hour. Wht is the speed in meters per second?. A hose crries pressure of 00 pounds per squre inch. Wht is the pressure in kilogrms per squre centimeter? 3. A concrete truck pours concrete t the rte of cubic rd per minute. Wht is the rte in cubic feet per hour?. Wter in pipe flows t rte of 0 gllons per minute. Wht is the rte in liters per second? 38 Chpter 7 Rtionl Functions
7. Inverse Vrition Essentil Question How cn ou recognize when two quntities vr directl or inversel? Recognizing Direct Vrition Work with prtner. You hng different weights from the sme spring. equilibrium REASONING QUANTITATIVELY To be proficient in mth, ou need to mke sense of quntities nd their reltionships in problem situtions. 0 kg 0. kg 0. kg 0.3 kg. Describe the 0. kg reltionship between the weight nd the distnce d 0. kg the spring stretches from equilibrium. Eplin wh the distnce is sid to vr 0. kg directl with the weight. b. Estimte the vlues of d from the figure. Then drw 0.7 kg sctter plot of the dt. Wht re the chrcteristics of the grph? c. Write n eqution tht represents d s function of. d. In phsics, the reltionship between d nd is described b Hooke s Lw. How would ou describe Hooke s Lw? centimeters Recognizing Inverse Vrition 8 3 Work with prtner. The tble shows the length (in inches) nd the width (in inches) of rectngle. The re of ech rectngle is squre inches. in.. Cop nd complete the tble. b. Describe the reltionship between nd. Eplin wh is sid to vr inversel with. c. Drw sctter plot of the dt. Wht re the chrcteristics of the grph? d. Write n eqution tht represents s function of. Communicte Your Answer 3. How cn ou recognize when two quntities vr directl or inversel?. Does the flpping rte of the wings of bird vr directl or inversel with the length of its wings? Eplin our resoning. Section 7. Inverse Vrition 39
7. Lesson Wht You Will Lern Core Vocbulr inverse vrition, p. 30 constnt of vrition, p. 30 Previous direct vrition rtios Clssif direct nd inverse vrition. Write inverse vrition equtions. Clssifing Direct nd Inverse Vrition You hve lerned tht two vribles nd show direct vrition when = for some nonzero constnt. Another tpe of vrition is clled inverse vrition. Core Concept Inverse Vrition Two vribles nd show inverse vrition when the re relted s follows: =, 0 The constnt is the constnt of vrition, nd is sid to vr inversel with. Clssifing Equtions Tell whether nd show direct vrition, inverse vrition, or neither.. = b. = STUDY TIP The eqution in prt (b) does not show direct vrition becuse = is not of the form =. c. = Given Eqution Solved for Tpe of Vrition. = = inverse b. = = neither c. = = direct Monitoring Progress Help in English nd Spnish t BigIdesMth.com Tell whether nd show direct vrition, inverse vrition, or neither.. =. = 0. 3. + = 0 The generl eqution = for direct vrition cn be rewritten s =. So, set of dt pirs (, ) shows direct vrition when the rtios re constnt. The generl eqution = for inverse vrition cn be rewritten s =. So, set of dt pirs (, ) shows inverse vrition when the products re constnt. 30 Chpter 7 Rtionl Functions
Clssifing Dt Tell whether nd show direct vrition, inverse vrition, or neither.. b. 8 3 3 8. Find the products nd rtios. = = 3 = 3 3 8 The products re constnt. The rtios re not constnt. ANALYZING RELATIONSHIPS In Emple (b), notice in the originl tble tht s increses b, is multiplied b. So, the dt in the tble represent n eponentil function. So, nd show inverse vrition. b. Find the products nd rtios. 8 = = 8 3 = So, nd show neither direct nor inverse vrition. The products re not constnt. The rtios re not constnt. Monitoring Progress Help in English nd Spnish t BigIdesMth.com Tell whether nd show direct vrition, inverse vrition, or neither.. 3. 3 0 0 0 30 0 Writing Inverse Vrition Equtions ANOTHER WAY Becuse nd vr inversel, ou lso know tht the products re constnt. This product equls the constnt of vrition. So, ou cn quickl determine tht = = 3() =. Writing n Inverse Vrition Eqution The vribles nd vr inversel, nd = when = 3. Write n eqution tht reltes nd. Then find when =. = = 3 Write generl eqution for inverse vrition. Substitute for nd 3 for. = Multipl ech side b 3. The inverse vrition eqution is =. When =, = =. Section 7. Inverse Vrition 3
Modeling with Mthemtics The time t (in hours) tht it tkes group of volunteers to build plground vries inversel with the number n of volunteers. It tkes group of 0 volunteers 8 hours to build the plground. Mke tble showing the time tht it would tke to build the plground when the number of volunteers is, 0,, nd 30. Wht hppens to the time it tkes to build the plground s the number of volunteers increses? LOOKING FOR A PATTERN Notice tht s the number of volunteers increses b, the time decreses b lesser nd lesser mount. From n = to n = 0, t decreses b hour 0 minutes. From n = 0 to n =, t decreses b 8 minutes. From n = to n = 30, t decreses b 3 minutes.. Understnd the Problem You re given description of two quntities tht vr inversel nd one pir of dt vlues. You re sked to crete tble tht gives dditionl dt pirs.. Mke Pln Use the time tht it tkes 0 volunteers to build the plground to find the constnt of vrition. Then write n inverse vrition eqution nd substitute for the different numbers of volunteers to find the corresponding times. 3. Solve the Problem t = n 8 = 0 Write generl eqution for inverse vrition. Substitute 8 for t nd 0 for n. 80 = Multipl ech side b 0. The inverse vrition eqution is t = 80. Mke tble of vlues. n n 0 30 t 80 = h 0 min 80 0 = h 80 = 3 h min 80 30 = h 0 min As the number of volunteers increses, the time it tkes to build the plground decreses.. Look Bck Becuse the time decreses s the number of volunteers increses, the time for volunteers to build the plground should be greter thn 8 hours. Monitoring Progress t = 80 = hours Help in English nd Spnish t BigIdesMth.com The vribles nd vr inversel. Use the given vlues to write n eqution relting nd. Then find when =.. =, = 7. =, = 8. =, = 9. WHAT IF? In Emple, it tkes group of 0 volunteers hours to build the plground. How long would it tke group of volunteers? 3 Chpter 7 Rtionl Functions
7. Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check. VOCABULARY Eplin how direct vrition equtions nd inverse vrition equtions re different.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both nswers. Wht is n inverse vrition eqution relting nd with =? Wht is n eqution for which the rtios re constnt nd =? Wht is n eqution for which vries inversel with nd =? Wht is n eqution for which the products re constnt nd =? Monitoring Progress nd Modeling with Mthemtics In Eercises 3 0, tell whether nd show direct vrition, inverse vrition, or neither. (See Emple.) 3. =. = 8. =. = 7. = + 8. + = 9. 8 = 0. = In Eercises, tell whether nd show direct vrition, inverse vrition, or neither. (See Emple.).. 8 3 9 3 3 98 3 39 37.. 7. 0 3.. 3 7. 90 In Eercises, the vribles nd vr inversel. Use the given vlues to write n eqution relting nd. Then find when = 3. (See Emple 3.). =, =. =, = 9 7. = 3, = 8 8. = 7, = 9. = 3, = 8 0. =, =. =, =. = 3, = 7 ERROR ANALYSIS In Eercises 3 nd, the vribles nd vr inversel. Describe nd correct the error in writing n eqution relting nd. 3. = 8, = = = (8) = 8 So, =. 8 3.. 8 8. 0. 0 7. 7 0 9. =, = = = 0 = So, = 0. Section 7. Inverse Vrition 33
. MODELING WITH MATHEMATICS The number of songs tht cn be stored on n MP3 pler vries inversel with the verge size of song. A certin MP3 pler cn store 00 songs when the verge size of song is megbtes (MB). (See Emple.). Mke tble showing the numbers of songs tht will fit on the MP3 pler when the verge size of song is MB,. MB, 3 MB, nd MB. b. Wht hppens to the number of songs s the verge song size increses?. MODELING WITH MATHEMATICS When ou stnd on snow, the verge pressure P (in pounds per squre inch) tht ou eert on the snow vries inversel with the totl re A (in squre inches) of the soles of our footwer. Suppose the pressure is 0.3 pound per squre inch when ou wer the snowshoes shown. Write n eqution tht gives P s function of A. Then find the pressure when ou wer the boots shown. Snowshoes: A = 30 in. 8. HOW DO YOU SEE IT? Does the grph of f represent inverse vrition or direct vrition? Eplin our resoning. 9. MAKING AN ARGUMENT You hve enough mone to bu hts for $0 ech or 0 hts for $ ech. Your friend ss this sitution represents inverse vrition. Is our friend correct? Eplin our resoning. 30. THOUGHT PROVOKING The weight w (in pounds) of n object vries inversel with the squre of the distnce d (in miles) of the object from the center of Erth. At se level (3978 miles from the center of the Erth), n stronut weighs 0 pounds. How much does the stronut weigh 00 miles bove se level? f Boots: A = 0 in. 7. PROBLEM SOLVING Computer chips re etched onto silicon wfers. The tble compres the re A (in squre millimeters) of computer chip with the number c of chips tht cn be obtined from silicon wfer. Write model tht gives c s function of A. Then predict the number of chips per wfer when the re of chip is 8 squre millimeters. Are (mm ), A 8 70 Number of chips, c 8 39 37 3. OPEN-ENDED Describe rel-life sitution tht cn be modeled b n inverse vrition eqution. 3. CRITICAL THINKING Suppose vries inversel with nd vries inversel with z. How does vr with z? Justif our nswer. 33. USING STRUCTURE To blnce the bord in the digrm, the distnce (in feet) of ech niml from the center of the bord must vr inversel with its weight (in pounds). Wht is the distnce of ech niml from the fulcrum? Justif our nswer. d ft ft Mintining Mthemticl Proficienc Divide. (Section.3) 7 lb fulcrum lb Reviewing wht ou lerned in previous grdes nd lessons 3. ( + 99) ( + ) 3. (3 3 3 + 30) (3 + ) Grph the function. Then stte the domin nd rnge. (Section.) 3. f() = + 37. g() = e 38. = ln 3 39. h() = ln ( + 9) 3 Chpter 7 Rtionl Functions
7. Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions with liner numertor nd liner denomintor is f() =. Prent function The grph of this function, shown t the right, is hperbol. Identifing Grphs of Rtionl Functions Work with prtner. Ech function is trnsformtion of the grph of the prent function f() =. Mtch the function with its grph. Eplin our resoning. Then describe the trnsformtion.. g() = b. g() = c. g() = + d. g() = e. g() = f. g() = + + + A. B. C. D. E. F. LOOKING FOR STRUCTURE To be proficient in mth, ou need to look closel to discern pttern or structure. Communicte Your Answer. Wht re some of the chrcteristics of the grph of rtionl function? 3. Determine the intercepts, smptotes, domin, nd rnge of the rtionl function g() =. b Section 7. Grphing Rtionl Functions 3
7. Lesson Wht You Will Lern Core Vocbulr rtionl function, p. 3 Previous domin rnge smptote long division Grph simple rtionl functions. Trnslte simple rtionl functions. Grph other rtionl functions. Grphing Simple Rtionl Functions A rtionl function hs the form f() = p(), where p() nd q() re polnomils q() nd q() 0. The inverse vrition function f() = is rtionl function. The grph of this function when = is shown below. Core Concept Prent Function for Simple Rtionl Functions STUDY TIP Notice tht 0 s nd s. This eplins wh = 0 is horizontl smptote of the grph of f() =. You cn lso nlze -vlues s pproches 0 to see wh = 0 is verticl smptote. LOOKING FOR STRUCTURE Becuse the function is of the form g() = f(), where =, the grph of g is verticl stretch b fctor of of the grph of f. The grph of the prent function f() = is hperbol, which consists of two smmetricl prts clled brnches. The domin nd rnge re ll nonzero rel numbers. An function of the form g() = ( 0) hs the sme smptotes, domin, nd rnge s the function f() =. Grphing Rtionl Function of the Form = Grph g() =. Compre the grph with the grph of f() =. Step The function is of the form g() =, so the smptotes re = 0 nd = 0. Drw the smptotes. Step Mke tble of vlues nd plot the points. Include both positive nd negtive vlues of. 3 3 3 3 Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. verticl smptote = 0 f() = horizontl smptote = 0 The grph of g lies frther from the es thn the grph of f. Both grphs lie in the first nd third qudrnts nd hve the sme smptotes, domin, nd rnge. f g Monitoring Progress Help in English nd Spnish t BigIdesMth.com. Grph g() =. Compre the grph with the grph of f() =. 3 Chpter 7 Rtionl Functions
Trnslting Simple Rtionl Functions Core Concept Grphing Trnsltions of Simple Rtionl Functions To grph rtionl function of the form = + k, follow these steps: h Step Drw the smptotes = h nd = k. = + k h Step Plot points to the left nd to the right of the verticl smptote. = k Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. = h Grphing Trnsltion of Rtionl Function Grph g() =. Stte the domin nd rnge. + LOOKING FOR STRUCTURE Let f() =. Notice tht g is of the form g() = f( h) + k, where h = nd k =. So, the grph of g is trnsltion units left nd unit down of the grph of f. Step Drw the smptotes = nd =. Step Plot points to the left of the verticl smptote, such s ( 3, 3), (, ), nd (, 0). Plot points to the right of the verticl smptote, such s (, ), (0, 3), nd (, ). Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. ( 3, 3) (, ) (, 0) The domin is ll rel numbers ecept nd the rnge is ll rel numbers ecept. (, ) (, ) (0, 3) Monitoring Progress Grph the function. Stte the domin nd rnge. Help in English nd Spnish t BigIdesMth.com. = 3 3. = +. = + Grphing Other Rtionl Functions All rtionl functions of the form = + b lso hve grphs tht re hperbols. c + d The verticl smptote of the grph is the line = d becuse the function is c undefined when the denomintor c + d is zero. The horizontl smptote is the line = c. Section 7. Grphing Rtionl Functions 37
Grphing Rtionl Function of the Form = + b c + d Grph f() = +. Stte the domin nd rnge. 3 (, 3 ) 8 ( ) 0, 3 (, 9) (, 3 3 ) 8, 7 ( ) 8 (, ) Step Drw the smptotes. Solve 3 = 0 for to find the verticl smptote = 3. The horizontl smptote is the line = c = =. Step Plot points to the left of the verticl smptote, such s (, ), ( 0, 3 ), nd (, 3 ). Plot points to the right of the verticl smptote, such s (, 9), (, 3 3 ), nd ( 8, 7 ). Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. The domin is ll rel numbers ecept 3 nd the rnge is ll rel numbers ecept. Rewriting rtionl function m revel properties of the function nd its grph. For emple, rewriting rtionl function in the form = + k revels tht it is h trnsltion of = with verticl smptote = h nd horizontl smptote = k. Rewriting nd Grphing Rtionl Function ANOTHER WAY You will use different method to rewrite g in Emple of Lesson 7.. Rewrite g() = 3 + in the form g() = + k. Grph the function. Describe + h the grph of g s trnsformtion of the grph of f() =. Rewrite the function 3 b using long division: + ) 3 + 3 + 3 The rewritten function is g() = + + 3. The grph of g is trnsltion unit left g nd 3 units up of the grph of f() =. 38 Chpter 7 Rtionl Functions Monitoring Progress Grph the function. Stte the domin nd rnge.. f() = + 3 8. Rewrite g() = + 3 + Help in English nd Spnish t BigIdesMth.com. f() = + 7. f() = 3 + in the form g() = + k. Grph the function. h Describe the grph of g s trnsformtion of the grph of f() =.
Modeling with Mthemtics A 3-D printer builds up lers of mterils to mke three-dimensionl models. Ech deposited ler bonds to the ler below it. A compn decides to mke smll displ models of engine components using 3-D printer. The printer costs $000. The mteril for ech model costs $0. Estimte how mn models must be printed for the verge cost per model to fll to $90. Wht hppens to the verge cost s more models re printed? USING A GRAPHING CALCULATOR Becuse the number of models nd verge cost cnnot be negtive, choose viewing window in the first qudrnt.. Understnd the Problem You re given the cost of printer nd the cost to crete model using the printer. You re sked to find the number of models for which the verge cost flls to $90.. Mke Pln Write n eqution tht represents the verge cost. Use grphing clcultor to estimte the number of models for which the verge cost is bout $90. Then nlze the horizontl smptote of the grph to determine wht hppens to the verge cost s more models re printed. 3. Solve the Problem Let c be the verge cost (in dollrs) nd m be the number of models printed. (Unit cost)(number printed) + (Cost of printer) 0m + 000 c = = Number printed m Use grphing clcultor to grph the function. Using the trce feture, the verge cost flls to $90 per model fter bout models re printed. Becuse the horizontl smptote is c = 0, the verge cost pproches $0 s more models re printed.. Look Bck Use grphing clcultor to crete tbles of vlues for lrge vlues of m. The tbles show tht the verge cost pproches $0 s more models re printed. 00 c = 0m + 000 m 0 X=.0383 Y=89.83008 0 0 X 0 00 0 00 0 300 X=0 Y ERROR 70 0.7 3.333 X 0000 0000 30000 0000 0000 0000 X=0 Y ERROR 0. 0.0 0.033 0.0 0.0 0.07 Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. WHAT IF? How do the nswers in Emple chnge when the cost of the 3-D printer is $800? Section 7. Grphing Rtionl Functions 39
7. Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check 7. COMPLETE THE SENTENCE The function = + 3 hs (n) of ll rel numbers + ecept 3 nd (n) of ll rel numbers ecept.. WRITING Is f() = 3 + rtionl function? Eplin our resoning. + Monitoring Progress nd Modeling with Mthemtics In Eercises 3 0, grph the function. Compre the grph with the grph of f() =. (See Emple.) 3. g() = 3. g() = 0 0. =. g() =. g() = 9 7. g() = 8. g() = 3 9. g() = 0. 0. g() = 0. In Eercises 8, grph the function. Stte the domin nd rnge. (See Emple.). g() = + 3. = 3 ANALYZING RELATIONSHIPS In Eercises, mtch the function with its grph. Eplin our resoning.. g() = 3 +. h() = + 3 + 3. h() =. = + 3. h() =. f() = + 7 3 7. g() = 8. = 0 + 7 3. f() = 3. = + 3 A. B. ERROR ANALYSIS In Eercises 9 nd 0, describe nd correct the error in grphing the rtionl function. 9. = 8 C. D. 8 370 Chpter 7 Rtionl Functions
In Eercises 3, grph the function. Stte the domin nd rnge. (See Emple 3.). f() = + 3. = +. REASONING Wht re the -intercept(s) of the grph of the function =? A, B 7. = + 8 9. f() = + + 3. h() = 3 8. h() = 8 + 3 30. g() = 3 3. = + 3 + 0 C D. USING TOOLS The time t (in seconds) it tkes for sound to trvel kilometer cn be modeled b 000 t = 0.T + 33 where T is the ir temperture (in degrees Celsius). In Eercises 33 0, rewrite the function in the form g() = + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f() =. (See Emple.) 33. g() = + + 3. g() = 37. g() = + 8 7 0 39. g() = + 3 3. g() = 7 + 3 3. g() = 38. g() = + 8 0. g() = 9 3 + 7. You re kilometer from lightning strike. You her the thunder.9 seconds lter. Use grph to find the pproimte ir temperture. b. Find the verge rte of chnge in the time it tkes sound to trvel kilometer s the ir temperture increses from 0 C to 0 C.. PROBLEM SOLVING Your school purchses mth softwre progrm. The progrm hs n initil cost of $00 plus $0 for ech student tht uses the progrm. (See Emple.). Estimte how mn students must use the progrm for the verge cost per student to fll to $30. b. Wht hppens to the verge cost s more students use the progrm?. PROBLEM SOLVING To join rock climbing gm, ou must p n initil fee of $00 nd monthl fee of $9.. Estimte how mn months ou must purchse membership for the verge cost per month to fll to $9. b. Wht hppens to the verge cost s the number of months tht ou re member increses? 3. USING STRUCTURE Wht is the verticl smptote of the grph of the function = + + 7? A = 7 B = C = D = 7. MODELING WITH MATHEMATICS A business is studing the cost to remove pollutnt from the ground t its site. The function =. models the estimted cost (in thousnds of dollrs) to remove percent (epressed s deciml) of the pollutnt.. Grph the function. Describe resonble domin nd rnge. b. How much does it cost to remove 0% of the pollutnt? 0% of the pollutnt? 80% of the pollutnt? Does doubling the percentge of the pollutnt removed double the cost? Eplin. USING TOOLS In Eercises 7 0, use grphing clcultor to grph the function. Then determine whether the function is even, odd, or neither. 7. h() = + 3 8. f() = 9 9. = 3 + 0. f() = 3 Section 7. Grphing Rtionl Functions 37
. MAKING AN ARGUMENT Your friend clims it is. ABSTRACT REASONING Describe the intervls where the grph of = is incresing or decresing when () > 0 nd (b) < 0. Eplin our resoning. possible for rtionl function to hve two verticl smptotes. Is our friend correct? Justif our nswer. 7. PROBLEM SOLVING An Internet service provider. HOW DO YOU SEE IT? Use the grph of f to chrges $0 instlltion fee nd monthl fee of $3. The tble shows the verge monthl costs of competing provider for months of service. Under wht conditions would person choose one provider over the other? Eplin our resoning. determine the equtions of the smptotes. Eplin. f 8 Months, Averge monthl cost (dollrs), $9.83 $.9 8 $.9 $. 3. DRAWING CONCLUSIONS In wht line(s) is the grph of = smmetric? Wht does this smmetr tell ou bout the inverse of the function f() =? 8. MODELING WITH MATHEMATICS The Doppler effect occurs when the source of sound is moving reltive to listener, so tht the frequenc fℓ(in hertz) herd b the listener is different from the frequenc fs (in hertz) t the source. In both equtions below, r is the speed (in miles per hour) of the sound source.. THOUGHT PROVOKING There re four bsic tpes of conic sections: prbol, circle, ellipse, nd hperbol. Ech of these cn be represented b the intersection of double-npped cone nd plne. The intersections for prbol, circle, nd ellipse re shown below. Sketch the intersection for hperbol. Moving w: 70fs f= 70 + r Prbol Circle Approching: 70fs f= 70 r. An mbulnce siren hs frequenc of 000 hertz. Write two equtions modeling the frequencies herd when the mbulnce is pproching nd when the mbulnce is moving w. Ellipse b. Grph the equtions in prt () using the domin 0 r 0.. REASONING The grph of the rtionl function f is hperbol. The smptotes of the grph of f intersect t (3, ). The point (, ) is on the grph. Find nother point on the grph. Eplin our resoning. c. For n speed r, how does the frequenc herd for n pproching sound source compre with the frequenc herd when the source moves w? Mintining Mthemticl Proficienc Reviewing wht ou lerned in previous grdes nd lessons Fctor the polnomil. (Skills Review Hndbook) 9. 80 0. 3 3.. 0 + 3 Simplif the epression. (Section.) 3. 3 3 37 Chpter 7 hsnb_lg_pe_070.indd 37. / 3/ /. / 8. 0 Rtionl Functions // :3 PM
7. 7. Wht Did You Lern? Core Vocbulr inverse vrition, p. 30 constnt of vrition, p. 30 rtionl function, p. 3 Core Concepts Section 7. Inverse Vrition, p. 30 Writing Inverse Vrition Equtions, p. 3 Section 7. Prent Function for Simple Rtionl Functions, p. 3 Grphing Trnsltions of Simple Rtionl Functions, p. 37 Mthemticl Prctices. Eplin the mening of the given informtion in Eercise on pge 3.. How re ou ble to recognize whether the logic used in Eercise 9 on pge 3 is correct or flwed? 3. How cn ou evlute the resonbleness of our nswer in prt (b) of Eercise on pge 37?. How did the contet llow ou to determine resonble domin nd rnge for the function in Eercise on pge 37? Stud Skills Anlzing Your Errors Stud Errors Wht Hppens: You do not stud the right mteril or ou do not lern it well enough to remember it on test without resources such s notes. How to Avoid This Error: Tke prctice test. Work with stud group. Discuss the topics on the test with our techer. Do not tr to lern whole chpter s worth of mteril in one night. 373
7. 7. Quiz Tell whether nd show direct vrition, inverse vrition, or neither. Eplin our resoning. (Section 7.). + = 7. = 3. = 0.. 3 9. 3. 8 9 8 7 3 8 7 3 8 9 7. The vribles nd vr inversel, nd = 0 when =. Write n eqution tht reltes nd. Then find when =. (Section 7.) Mtch the eqution with the correct grph. Eplin our resoning. (Section 7.) 8. f() = 3 + 9. = + 3 0. h() = + 3 + A. B. C.. Rewrite g() = + 9 + 8 in the form g() = + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f() =. (Section 7.). The time t (in minutes) required to empt tnk vries inversel with the pumping rte r (in gllons per minute). The rte of certin pump is 70 gllons per minute. It tkes the pump 0 minutes to empt the tnk. Complete the tble for the times it tkes the pump to empt tnk for the given pumping rtes. (Section 7.) Pumping rte (gl/min) 0 0 Time (min) 3. A pitcher throws strikes in the first 38 pitches. The tble shows how pitcher s strike percentge chnges when the pitcher throws consecutive strikes fter the first 38 pitches. Write rtionl function for the strike percentge in terms of. Grph the function. How mn consecutive strikes must the pitcher throw to rech strike percentge of 0.0? (Section 7.) Totl strikes Totl pitches Strike percentge 0 38 0. 3 0.9 0 8 0. + + 38 70 37 Chpter 7 Rtionl Functions
7.3 Multipling nd Dividing Rtionl Epressions Essentil Question How cn ou determine the ecluded vlues in product or quotient of two rtionl epressions? You cn multipl nd divide rtionl epressions in much the sme w tht ou multipl nd divide frctions. Vlues tht mke the denomintor of n epression zero re ecluded vlues. + = +, 0 + = + = +, Product of rtionl epressions Quotient of rtionl epressions Multipling nd Dividing Rtionl Epressions REASONING ABSTRACTLY To be proficient in mth, ou need to know nd fleibl use different properties of opertions nd objects. Work with prtner. Find the product or quotient of the two rtionl epressions. Then mtch the product or quotient with its ecluded vlues. Eplin our resoning. Product or Quotient Ecluded Vlues. b. c. + + d. + + e. f. g. + + + + + h. + = = A., 0, nd = B. nd = C., 0, nd = D. nd = E., 0, nd = F. nd + = H. G. nd Writing Product or Quotient Work with prtner. Write product or quotient of rtionl epressions tht hs the given ecluded vlues. Justif our nswer.. b. nd 3 c., 0, nd 3 Communicte Your Answer 3. How cn ou determine the ecluded vlues in product or quotient of two rtionl epressions?. Is it possible for the product or quotient of two rtionl epressions to hve no ecluded vlues? Eplin our resoning. If it is possible, give n emple. Section 7.3 Multipling nd Dividing Rtionl Epressions 37
7.3 Lesson Wht You Will Lern Core Vocbulr rtionl epression, p. 37 simplified form of rtionl epression, p. 37 Previous frctions polnomils domin equivlent epressions reciprocl Simplif rtionl epressions. Multipl rtionl epressions. Divide rtionl epressions. Simplifing Rtionl Epressions A rtionl epression is frction whose numertor nd denomintor re nonzero polnomils. The domin of rtionl epression ecludes vlues tht mke the denomintor zero. A rtionl epression is in simplified form when its numertor nd denomintor hve no common fctors (other thn ±). Core Concept Simplifing Rtionl Epressions Let, b, nd c be epressions with b 0 nd c 0. Propert c bc = b Divide out common fctor c. STUDY TIP Notice tht ou cn divide out common fctors in the second epression t the right. You cnnot, however, divide out like terms in the third epression. Emples = 3 3 = 3 3 ( + 3) ( + 3)( + 3) = + 3 Simplifing rtionl epression usull requires two steps. First, fctor the numertor nd denomintor. Then, divide out n fctors tht re common to both the numertor nd denomintor. Here is n emple: + 7 = ( + 7) Divide out common fctor. Divide out common fctor + 3. = + 7 Simplifing Rtionl Epression Simplif. COMMON ERROR Do not divide out vrible terms tht re not fctors. ( + )( ) = ( + )( ) ( + )( ) = ( + )( ) Fctor numertor nd denomintor. Divide out common fctor. =, Simplified form The originl epression is undefined when =. To mke the originl nd simplified epressions equivlent, restrict the domin of the simplified epression b ecluding =. Both epressions re undefined when =, so it is not necessr to list it. 37 Chpter 7 Rtionl Functions Monitoring Progress Simplif the rtionl epression, if possible.. ( + ) ( + )( + 3). + Help in English nd Spnish t BigIdesMth.com 3. ( + ). 3
ANOTHER WAY In Emple, ou cn first simplif ech rtionl epression, then multipl, nd finll simplif the result. 8 3 7 3 = 7 = 7 = 7, 0, 0 Multipling Rtionl Epressions The rule for multipling rtionl epressions is the sme s the rule for multipling numericl frctions: multipl numertors, multipl denomintors, nd write the new frction in simplified form. Similr to rtionl numbers, rtionl epressions re closed under multipliction. Core Concept Multipling Rtionl Epressions Let, b, c, nd d be epressions with b 0 nd d 0. Propert Emple Find the product 83 7 3. b c d = c bd 3 0 = 303 3 8 3 7 3 = 7 8 3 Multipling Rtionl Epressions = 8 7 3 8 Simplif c if possible. bd 0 3 = 0 3 3 0 3 = 3 Multipl numertors nd denomintors. 3 Fctor nd divide out common fctors. = 7, 0, 0 Simplified form, 0, 0 Multipling Rtionl Epressions 3 3 Find the product + + 0. 3 3 3 + + 0 3 3( ) ( + )( ) = ( )( + ) 3 Fctor numertors nd denomintors. 3( )( + )( ) = ( )( + )(3) Multipl numertors nd denomintors. 3( )( )( + )( ) = ( )( + )(3) Rewrite s ( )( ). Check X - -3 - - 0 X=- Y ERROR 8 7 ERROR ERROR 9 8 7 3 Y 3( )( )( + )( ) = ( )( + )(3) Divide out common fctors. = +,, 0, Simplified form Check the simplified epression. Enter the originl epression s nd the simplified epression s in grphing clcultor. Then use the tble feture to compre the vlues of the two epressions. The vlues of nd re the sme, ecept when =, = 0, nd =. So, when these vlues re ecluded from the domin of the simplified epression, it is equivlent to the originl epression. Section 7.3 Multipling nd Dividing Rtionl Epressions 377
Multipling Rtionl Epression b Polnomil STUDY TIP Notice tht + 3 + 9 does not equl zero for n rel vlue of. So, no vlues must be ecluded from the domin to mke the simplified form equivlent to the originl. Find the product + 3 7 ( + 3 + 9). + 3 7 ( + 3 + 9) = + 3 7 + 3 + 9 = ( + )( + 3 + 9) ( 3)( + 3 + 9) = ( + )( + 3 + 9) ( 3)( + 3 + 9) = + 3 Write polnomil s rtionl epression. Multipl. Fctor denomintor. Divide out common fctors. Simplified form Monitoring Progress Find the product.. 3 8 9 3 Help in English nd Spnish t BigIdesMth.com. 0 + 3 7. + 3 ( + + ) Dividing Rtionl Epressions To divide one rtionl epression b nother, multipl the first rtionl epression b the reciprocl of the second rtionl epression. Rtionl epressions re closed under nonzero division. Core Concept Dividing Rtionl Epressions Let, b, c, nd d be epressions with b 0, c 0, nd d 0. Propert b c d = b d c = d bc Simplif d if possible. bc Emple 7 + + 3 = 7 + 3 + = 7( 3) ( + )( + ), 3 Dividing Rtionl Epressions 7 Find the quotient 0 + 30. 7 0 + 30 = 7 0 + 30 7 ( )( ) = ( ) ( ) 7( )( ) = ( )()( ) Multipl b reciprocl. Fctor. Multipl. Divide out common fctors. 378 Chpter 7 Rtionl Functions = 7, 0,, Simplified form
Dividing Rtionl Epression b Polnomil Find the quotient + (3 + ). + (3 + ) = + 3 + (3 + )( 3) = (3 + ) Fctor. (3 + )( 3) = ()(3 + ) = 3 3, 3 Solving Rel-Life Problem Multipl b reciprocl. Divide out common fctors. Simplified form The totl nnul mount I (in millions of dollrs) of personl income erned in Albm nd its nnul popultion P (in millions) cn be modeled b 9t + 0,97 I = 0.003t + nd P = 0.033t +.3 where t represents the er, with t = corresponding to 00. Find model M for the nnul per cpit income. (Per cpit mens per person.) Estimte the per cpit income in 00. (Assume t > 0.) To find model M for the nnul per cpit income, divide the totl mount I b the popultion P. 9t + 0,97 M = (0.033t +.3) Divide I b P. 0.003t + 9t + 0,97 = 0.003t + 0.033t +.3 9t + 0,97 = (0.003t + )(0.033t +.3) Multipl b reciprocl. Multipl. To estimte Albm s per cpit income in 00, let t = 0 in the model. 9 M = 0 + 0,97 (0.003 0 + )(0.033 0 +.3) 3,707 Substitute 0 for t. Use clcultor. In 00, the per cpit income in Albm ws bout $3,707. Monitoring Progress Find the quotient. 8. 0 + 8 Help in English nd Spnish t BigIdesMth.com 9. + 3 ( + ) Section 7.3 Multipling nd Dividing Rtionl Epressions 379
7.3 Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check. WRITING Describe how to multipl nd divide two rtionl epressions.. WHICH ONE DOESN T BELONG? Which rtionl epression does not belong with the other three? Eplin our resoning. + 9 + + 3 Monitoring Progress nd Modeling with Mthemtics In Eercises 3 0, simplif the epression, if possible. (See Emple.) 3. 3. 3 8 7 + 7. + + 8 3 + 8.. 8. 7 3 3 + 3 + 3 7 + 0 7 + 3 7 0. ( + 8). ERROR ANALYSIS Describe nd correct the error in simplifing the rtionl epression. 3 + + 8 + 8 + = + + 3 + + 9. 0. 3 0 3 + 3 3 3 + 7 7 7 7 In Eercises 0, find the product. (See Emples, 3, nd.). 3 8. 8 3 3. ERROR ANALYSIS Describe nd correct the error in finding the product. 3 3 ( + )( ) = + 3 3 + ( + )( )( 3) = (3 )( + ) =, 3, 3... 7. 8. 9. ( ) 3 3 ( + ) 9 ( 3)( + ) 3 ( 9)( + 8) 3 3 3 +. + 3 + + + + 3 3 + + + + 3 9 ( + 8) + 3 3. USING STRUCTURE Which rtionl epression is in simplified form? A C + 3 + + 9 3 B D + + 8 + 3 + 3 +. COMPARING METHODS Find the product below b multipling the numertors nd denomintors, then simplifing. Then find the product b simplifing ech epression, then multipling. Which method do ou prefer? Eplin. 3 380 Chpter 7 Rtionl Functions
. WRITING Compre the function (3 7)( + ) f() = to the function g() = +. (3 7). MODELING WITH MATHEMATICS Write model in terms of for the totl re of the bse of the building. 3 0 7 + 0 In Eercises 7 3, find the quotient. (See Emples nd.) 7. 9. 3. 3. 33. 3. 3 3 8 7 8 8. z 3 z 3 z 3 + 3 30. 7 + ( + + 9) 3 ( + 8 + 8) + 9 + 8 + + 8 3 8 + 8 3 0 + 8 0 + 3 + 0 + + 0 3 3 In Eercises 3 nd 3, use the following informtion. Mnufcturers often pckge products in w tht uses the lest mount of mteril. One mesure of the effi cienc of pckge is the rtio of its surfce re S to its volume V. The smller the rtio, the more effi cient the pckging. 3. PROBLEM SOLVING A popcorn compn is designing new tin with the sme squre bse nd twice the height of the old tin.. Write n epression for the efficienc rtio S V of ech tin. b. Did the compn mke good decision b creting the new tin? Eplin. 37. MODELING WITH MATHEMATICS The totl mount I (in millions of dollrs) of helthcre ependitures nd the residentil popultion P (in millions) in the United Sttes cn be modeled b 7,000t +,3,000 I = nd + 0.08t P =.9t + 78.9 where t is the number of ers since 000. Find model M for the nnul helthcre ependitures per resident. Estimte the nnul helthcre ependitures per resident in 00. (See Emple 7.) 38. MODELING WITH MATHEMATICS The totl mount I (in millions of dollrs) of school ependitures from prekindergrten to college level nd the enrollment P (in millions) in prekindergrten through college in the United Sttes cn be modeled b 7,93t + 709,9 I = nd P = 0.90t + 70.9 0.08t where t is the number of ers since 00. Find model M for the nnul eduction ependitures per student. Estimte the nnul eduction ependitures per student in 009. s s h s s h 3. You re emining three clindricl continers.. Write n epression for the efficienc rtio S V of clinder. b. Find the efficienc rtio for ech clindricl cn listed in the tble. Rnk the three cns ccording to efficienc. Soup Coffee Pint Height, h 0. cm.9 cm 9. cm Rdius, r 3. cm 7.8 cm 8. cm 39. USING EQUATIONS Refer to the popultion model P in Eercise 37.. Interpret the mening of the coefficient of t. b. Interpret the mening of the constnt term. Section 7.3 Multipling nd Dividing Rtionl Epressions 38
0. HOW DO YOU SEE IT? Use the grphs of f nd g to determine the ecluded vlues of the functions h() = ( fg)() nd k() = ( g) f (). Eplin our resoning. f g. CRITICAL THINKING Find the epression tht mkes the following sttement true. Assume nd. + 3 3 0 = + + 7 USING STRUCTURE In Eercises nd, perform the indicted opertions.. + ( + 9) 3. DRAWING CONCLUSIONS Complete the tble for the function = +. Then use the trce feture of grphing clcultor to eplin the behvior of the function t =. 3. 3.8 3.9... MAKING AN ARGUMENT You nd our friend re sked to stte the domin of the epression below. + 7 + Your friend clims the domin is ll rel numbers ecept. You clim the domin is ll rel numbers ecept 9 nd. Who is correct? Eplin. 3. MATHEMATICAL CONNECTIONS Find the rtio of the perimeter to the re of the tringle shown. 8 Mintining Mthemticl Proficienc Solve the eqution. Check our solution. (Skills Review Hndbook) 0.. ( 3 + 8) + 7. REASONING Animls tht live in tempertures severl degrees colder thn their bodies must void losing het to survive. Animls cn better conserve bod het s their surfce re to volume rtios decrese. Find the surfce re to volume rtio of ech penguin shown b using clinders to pproimte their shpes. Which penguin is better equipped to live in colder environment? Eplin our resoning. Glpgos Penguin 3 cm rdius = cm King Penguin 9 cm rdius = cm Not drwn to scle 8. THOUGHT PROVOKING Is it possible to write two rdicl functions whose product when grphed is prbol nd whose quotient when grphed is hperbol? Justif our nswer. 9. REASONING Find two rtionl functions f nd g tht hve the stted product nd quotient. g) (fg)() =, ( f () = ( ) ( + ) Reviewing wht ou lerned in previous grdes nd lessons + = 3 +. = 3 3. 3 = 9 3. + 3 = 3 Write the prime fctoriztion of the number. If the number is prime, then write prime. (Skills Review Hndbook).. 9. 7 7. 79 38 Chpter 7 Rtionl Functions
7. Adding nd Subtrcting Rtionl Epressions Essentil Question How cn ou determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme w tht ou dd nd subtrct frctions. + + + = + Sum of rtionl epressions + = = Difference of rtionl epressions Adding nd Subtrcting Rtionl Epressions Work with prtner. Find the sum or difference of the two rtionl epressions. Then mtch the sum or difference with its domin. Eplin our resoning. Sum or Difference Domin. b. c. d. e. + 3 = A. ll rel numbers ecept + = B. ll rel numbers ecept nd + = C. ll rel numbers ecept + = D. ll rel numbers ecept 0 + + = E. ll rel numbers ecept nd + + f. + = F. ll rel numbers ecept 0 nd CONSTRUCTING VIABLE ARGUMENTS To be proficient in mth, ou need to justif our conclusions nd communicte them to others. g. + = G. ll rel numbers ecept h. + + = H. ll rel numbers ecept 0 nd Writing Sum or Difference Work with prtner. Write sum or difference of rtionl epressions tht hs the given domin. Justif our nswer.. ll rel numbers ecept b. ll rel numbers ecept nd 3 c. ll rel numbers ecept, 0, nd 3 Communicte Your Answer 3. How cn ou determine the domin of the sum or difference of two rtionl epressions?. Your friend found sum s follows. Describe nd correct the error(s). + 3 = + 3 + Section 7. Adding nd Subtrcting Rtionl Epressions 383
7. Lesson Wht You Will Lern Core Vocbulr comple frction, p. 387 Previous rtionl numbers reciprocl Add or subtrct rtionl epressions. Rewrite rtionl epressions nd grph the relted function. Simplif comple frctions. Adding or Subtrcting Rtionl Epressions As with numericl frctions, the procedure used to dd (or subtrct) two rtionl epressions depends upon whether the epressions hve like or unlike denomintors. To dd (or subtrct) rtionl epressions with like denomintors, simpl dd (or subtrct) their numertors. Then plce the result over the common denomintor. Core Concept Adding or Subtrcting with Like Denomintors Let, b, nd c be epressions with c 0. Addition Subtrction c + b c = + b c c b c = b c Adding or Subtrcting with Like Denomintors. 7 + 3 = 7 + 3 b. = 0 = + + = + Add numertors nd simplif. Subtrct numertors. Monitoring Progress Find the sum or difference. 8.. 3 + 3 3. Help in English nd Spnish t BigIdesMth.com. + + + To dd (or subtrct) two rtionl epressions with unlike denomintors, find common denomintor. Rewrite ech rtionl epression using the common denomintor. Then dd (or subtrct). Core Concept Adding or Subtrcting with Unlike Denomintors Let, b, c, nd d be epressions with c 0 nd d 0. Addition Subtrction c + b d = d cd + bc cd = d + bc cd c b d = d cd bc cd = d bc cd 38 Chpter 7 Rtionl Functions You cn lws find common denomintor of two rtionl epressions b multipling the denomintors, s shown bove. However, when ou use the lest common denomintor (LCD), which is the lest common multiple (LCM) of the denomintors, simplifing our nswer m tke fewer steps.
To find the LCM of two (or more) epressions, fctor the epressions completel. The LCM is the product of the highest power of ech fctor tht ppers in n of the epressions. Finding Lest Common Multiple (LCM) Find the lest common multiple of nd +. Step Fctor ech polnomil. Write numericl fctors s products of primes. = ( ) = ( )( + )( ) + = ( + ) = ()(3)( ) Step The LCM is the product of the highest power of ech fctor tht ppers in either polnomil. LCM = ( )(3)( + )( ) = ( + )( ) Find the sum 7 9 + 3 + 3. Adding with Unlike Denomintors Method Use the definition for dding rtionl epressions with unlike denomintors. 7 9 + 3 + 3 = 7(3 + 3) + (9 ) 9 (3 + 3) = + + 9 3 9 (3 + 3) = 3(3 + 7 + 7) 9 ( + )(3) = 3 + 7 + 7 9 ( + ) c + b d = d + bc cd Distributive Propert Fctor. Divide out common fctors. Simplif. Method Find the LCD nd then dd. To find the LCD, fctor ech denomintor nd write ech fctor to the highest power tht ppers in either denomintor. Note tht 9 = 3 nd 3 + 3 = 3( + ), so the LCD is 9 ( + ). 7 9 + 3 + 3 = 7 9 + 3( + ) = 7 9 + + + 3( + ) 3 3 7 + 7 = 9 ( + ) + 3 9 ( + ) = 3 + 7 + 7 9 ( + ) Fctor second denomintor. LCD is 9 ( + ). Multipl. Add numertors. Note in Emples nd 3 tht when dding or subtrcting rtionl epressions, the result is rtionl epression. In generl, similr to rtionl numbers, rtionl epressions re closed under ddition nd subtrction. Section 7. Adding nd Subtrcting Rtionl Epressions 38
COMMON ERROR When subtrcting rtionl epressions, remember to distribute the negtive sign to ll the terms in the quntit tht is being subtrcted. Find the difference + + 3. Subtrcting with Unlike Denomintors + + 3 = + ( ) ( )( 3) + = ( ) 3 3 ( )( 3) = ( )( 3) ( )( 3) = ( ) ( )( 3) = + 3 ( )( 3) ( )( + ) = ( )( 3) Fctor ech denomintor. LCD is ( )( 3). Multipl. + =, Simplif. ( 3) Subtrct numertors. Simplif numertor. Fctor numertor. Divide out common fctors. Monitoring Progress Help in English nd Spnish t BigIdesMth.com. Find the lest common multiple of 3 nd 0. Find the sum or difference.. 3 7 7. 3 + 9 8. + 8 Rewriting Rtionl Functions Rewriting rtionl epression m revel properties of the relted function nd its grph. In Emple of Section 7., ou used long division to rewrite rtionl epression. In the net emple, ou will use inspection. Rewriting nd Grphing Rtionl Function Rewrite the function g() = 3 + in the form g() = + k. Grph the function. + h Describe the grph of g s trnsformtion of the grph of f() =. g Rewrite b inspection: 3 + + = 3 + 3 + 3( + ) + 3( + ) = = + + + + + = 3 + + The rewritten function is g() = + 3. The grph of g is trnsltion unit + left nd 3 units up of the grph of f() =. Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. Rewrite g() = in the form g() = + k. Grph the function. 3 h Describe the grph of g s trnsformtion of the grph of f() =. 38 Chpter 7 Rtionl Functions
Comple Frctions A comple frction is frction tht contins frction in its numertor or denomintor. A comple frction cn be simplified using either of the methods below. Core Concept Simplifing Comple Frctions Method If necessr, simplif the numertor nd denomintor b writing ech s single frction. Then divide b multipling the numertor b the reciprocl of the denomintor. Method Multipl the numertor nd the denomintor b the LCD of ever frction in the numertor nd denomintor. Then simplif. + Simplif + +. Method + + + Simplifing Comple Frction + = 3 + 8 ( + ) ( + ) = + 3 + 8 ( + ) = ( + )(3 + 8) = 3 + 8,, 0 Add frctions in denomintor. Multipl b reciprocl. Divide out common fctors. Simplif. Method The LCD of ll the frctions in the numertor nd denomintor is ( + ). + + + + = + + ( + ) ( + ) + ( + ) = + ( + ) + Divide ( + ) = = Monitoring Progress + ( + ) 3 + 8,, 0 Multipl numertor nd denomintor b the LCD. out common fctors. Simplif. Simplif. Help in English nd Spnish t BigIdesMth.com Simplif the comple frction. 3 0. 7.. 0 + 3 3 + 3 + + Section 7. Adding nd Subtrcting Rtionl Epressions 387
7. Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check. COMPLETE THE SENTENCE A frction tht contins frction in its numertor or denomintor is clled (n).. WRITING Eplin how dding nd subtrcting rtionl epressions is similr to dding nd subtrcting numericl frctions. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 8, find the sum or difference. (See Emple.) 3. +. 7. 9 + + + 3 + + 3.. 8. 3 8 + 8 In Eercises 9, find the lest common multiple of the epressions. (See Emple.) 9. 3, 3( ) 0., +., ( )., 8 3.,. 9, 3 +. + 3 0, 8. 3, + 7 ERROR ANALYSIS In Eercises 7 nd 8, describe nd correct the error in finding the sum. 7. 8. + = + + = ( + ) + + = + ( + )( ) In Eercises 9, find the sum or difference. (See Emples 3 nd.) 9.. 3. 7 3 + + + + 3 3 0.. 8 3 + 9 3 + +.. + + 3 + 7 + 3 + 3 + 3. + + + 3 REASONING In Eercises 7 nd 8, tell whether the sttement is lws, sometimes, or never true. Eplin. 7. The LCD of two rtionl epressions is the product of the denomintors. 8. The LCD of two rtionl epressions will hve degree greter thn or equl to tht of the denomintor with the higher degree. 9. ANALYZING EQUATIONS How would ou begin to rewrite the function g() = + to obtin the form + g() = h + k? ( + ) 7 A g() = + ( + ) + B g() = + ( + ) + (3 ) C g() = + D g() = + + 30. ANALYZING EQUATIONS How would ou begin to rewrite the function g() = to obtin the form g() = h + k? ( +)( ) A g() = B g() = + C g() = + D g() = 388 Chpter 7 Rtionl Functions
In Eercises 3 38, rewrite the function g in the form g() = + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f() =. (See Emple.) 7 3. g() = + + 3. g() = + 3 3. g() = 3. g() = two resistors in prllel circuit with resistnces R nd R (in ohms) is given b the eqution shown. Simplif the comple frction. Then find the totl resistnce when R = 000 ohms nd R = 00 ohms. Rt = + R R 8 + 3 33. g() = R 3. g() = 3 + 3 Rt 7 9 + 0 37. g() = = R 38. g() = In Eercises 39, simplif the comple frction. (See Emple.) 3 0. + 39. 0 +. REWRITING A FORMULA The totl resistnce Rt of 7 8 0.. + + 7. PROBLEM SOLVING You pln trip tht involves 0-mile bus ride nd trin ride. The entire trip is 0 miles. The time (in hours) the bus trvels is 0 =, where is the verge speed (in miles per hour) of the bus. The time (in hours) the trin trvels 00 is =. Write nd simplif model tht shows + 30 the totl time of the trip. 8. PROBLEM SOLVING You prticipte in sprint trithlon tht involves swimming, biccling, nd running. The tble shows the distnces (in miles) nd our verge speed for ech portion of the rce. 3 3 3 3.. 3 + + + + 3. PROBLEM SOLVING The totl time T (in hours) needed to fl from New York to Los Angeles nd bck cn be modeled b the eqution below, where d is the distnce (in miles) ech w, is the verge irplne speed (in miles per hour), nd j is the verge speed (in miles per hour) of the jet strem. Simplif the eqution. Then find the totl time it tkes to fl 8 miles when = 0 miles per hour nd j = miles per hour. d d T=+ j +j Distnce (miles) Speed (miles per hour) Swimming 0. r Biccling r Running r+. Write model in simplified form for the totl time (in hours) it tkes to complete the rce. b. How long does it tke to complete the rce if ou cn swim t n verge speed of miles per hour? Justif our nswer. 9. MAKING AN ARGUMENT Your friend clims tht NY LA j j NY LA A j +j Section 7. hsnb_lg_pe_070.indd 389 the lest common multiple of two numbers is lws greter thn ech of the numbers. Is our friend correct? Justif our nswer. Adding nd Subtrcting Rtionl Epressions 389 // : PM
0. HOW DO YOU SEE IT? Use the grph of the function f() = h + k to determine the vlues of h nd k.. REWRITING A FORMULA You borrow P dollrs to bu cr nd gree to rep the lon over t ers t monthl interest rte of i (epressed s deciml). Your monthl pment M is given b either formul below. Pi M = t ( + i) Pi( + i) or M = t ( + i) t. Show tht the formuls re equivlent b simplifing the first formul. b. Find our monthl pment when ou borrow $,00 t monthl interest rte of 0.% nd rep the lon over ers.. THOUGHT PROVOKING Is it possible to write two rtionl functions whose sum is qudrtic function? Justif our nswer. 3. USING TOOLS Use technolog to rewrite the (97.)(0.0) + (0.003) function g() = in the. + form g() = + k. Describe the grph of g s h trnsformtion of the grph of f() =.. MATHEMATICAL CONNECTIONS Find n epression for the surfce re of the bo. f + 3. PROBLEM SOLVING You re hired to wsh the new crs t cr delership with two other emploees. You tke n verge of 0 minutes to wsh cr (R = /0 cr per minute). The second emploee wshes cr in minutes. The third emploee wshes cr in + 0 minutes.. Write epressions for the rtes tht ech emploee cn wsh cr. b. Write single epression R for the combined rte of crs wshed per minute b the group. c. Evlute our epression in prt (b) when the second emploee wshes cr in 3 minutes. How mn crs per hour does this represent? Eplin our resoning.. MODELING WITH MATHEMATICS The mount A (in milligrms) of spirin in person s bloodstrem cn be modeled b 39t A = + 0. 0.8t + 0.99t + where t is the time (in hours) fter one dose is tken. A first dose second dose A combined effect. A second dose is tken hour fter the first dose. Write n eqution to model the mount of the second dose in the bloodstrem. b. Write model for the totl mount of spirin in the bloodstrem fter the second dose is tken. 7. FINDING A PATTERN Find the net two epressions in the pttern shown. Then simplif ll five epressions. Wht vlue do the epressions pproch? + +, +, +,... + + + + + + + Mintining Mthemticl Proficienc Solve the sstem b grphing. (Section 3.) Reviewing wht ou lerned in previous grdes nd lessons 8. = + 9. 3 = 0 0. 3 =. = ( + ) 3 = 3 + = 9 = 3 = + + 390 Chpter 7 Rtionl Functions
7. Solving Rtionl Equtions Essentil Question How cn ou solve rtionl eqution? Solving Rtionl Equtions Work with prtner. Mtch ech eqution with the grph of its relted sstem of equtions. Eplin our resoning. Then use the grph to solve the eqution.. d. = b. = e. = = c. 3 = + f. = A. B. C. D. E. F. MAKING SENSE OF PROBLEMS To be proficient in mth, ou need to pln solution pthw rther thn simpl jumping into solution ttempt. Solving Rtionl Equtions Work with prtner. Look bck t the equtions in Eplortions (d) nd (e). Suppose ou wnt more ccurte w to solve the equtions thn using grphicl pproch.. Show how ou could use numericl pproch b creting tble. For instnce, ou might use spredsheet to solve the equtions. b. Show how ou could use n nlticl pproch. For instnce, ou might use the method ou used to solve proportions. Communicte Your Answer 3. How cn ou solve rtionl eqution?. Use the method in either Eplortion or to solve ech eqution.. + = + b. + = + c. = Section 7. Solving Rtionl Equtions 39
7. Lesson Wht You Will Lern Core Vocbulr cross multipling, p. 39 Previous proportion etrneous solution inverse of function Solve rtionl equtions b cross multipling. Solve rtionl equtions b using the lest common denomintor. Use inverses of functions. Solving b Cross Multipling You cn use cross multipling to solve rtionl eqution when ech side of the eqution is single rtionl epression. 3 Solve + = 9 +. Solving Rtionl Eqution b Cross Multipling Check 3 + =? 9 ( ) + 3 =? 9 3 3 = 3 3 + = 9 + Write originl eqution. 3( + ) = 9( + ) Cross multipl. + = 9 + 9 3 + = 9 3 = Distributive Propert Subtrct 9 from ech side. Subtrct from ech side. = Divide ech side b 3. The solution is =. Check this in the originl eqution. 39 Chpter 7 Rtionl Functions Writing nd Using Rtionl Model An llo is formed b miing two or more metls. Sterling silver is n llo composed of 9.% silver nd 7.% copper b weight. You hve ounces of 800 grde silver, which is 80% silver nd 0% copper b weight. How much pure silver should ou mi with the 800 grde silver to mke sterling silver? percent of copper in miture = weight of copper in miture totl weight of miture 7. 00 = (0.)() + is the mount of silver dded. 7.( + ) = 00(0.)() Cross multipl.. + 7. = 300 Simplif. 7. = 87. Subtrct. from ech side. = Divide ech side b 7.. You should mi ounces of pure silver with the ounces of 800 grde silver. Monitoring Progress Help in English nd Spnish t BigIdesMth.com Solve the eqution b cross multipling. Check our solution(s).. 3 =. 7 + 3 = 3. 3 + = + 8
Solving b Using the Lest Common Denomintor When rtionl eqution is not epressed s proportion, ou cn solve it b multipling ech side of the eqution b the lest common denomintor of the rtionl epressions. Solving Rtionl Equtions b Using the LCD Solve ech eqution.. + 7 = 9 b. 8 = 3 Check 8 + 7 =? 9 8 8 + 8 =? 9 8 9 8 = 9 8. + 7 = 9 Write originl eqution. ) ( = 9 ) Multipl ech side b the LCD,. 0 + 7 = 3 Simplif. ( + 7 7 = Subtrct 0 from ech side. = 8 Divide ech side b 7. The solution is = 8. Check this in the originl eqution. 8 b. = 3 Write originl eqution. 8 ( ) ( ) = ( ) 3 ( ) 8 = 3( ) 8 = 3 + = 0 Multipl ech side b the LCD, ( ). Simplif. ( )( ) = 0 Fctor. = or = Distributive Propert Write in stndrd form. Zero-Product Propert The solutions re = nd =. Check these in the originl eqution. Check 8 =? 3 8 Substitute for. =? 3 + =? 3 Simplif. =? 3 = 3 = Monitoring Progress Help in English nd Spnish t BigIdesMth.com Solve the eqution b using the LCD. Check our solution(s).. + = 7. 3 + = 3. + + = + 3 Section 7. Solving Rtionl Equtions 393
When solving rtionl eqution, ou m obtin solutions tht re etrneous. Be sure to check for etrneous solutions b checking our solutions in the originl eqution. Solve 3 = 8 9 + 3. Solving n Eqution with n Etrneous Solution Write ech denomintor in fctored form. The LCD is ( + 3)( 3). 3 = 8 ( + 3)( 3) + 3 ( + 3)( 3) 3 = ( + 3)( 3) 8 ( + 3)( 3) ( + 3)( 3) + 3 ( + 3) = 8 ( 3) + 8 = 8 + 0 = + 8 0 = + 3 9 0 = ( 3)( + 3) 3 = 0 or + 3 = 0 = 3 or = 3 Check ANOTHER WAY You cn lso grph ech side of the eqution nd find the -vlue where the grphs intersect. 8 0 Check = 3 : Check = 3: = 3 3? 8 ( 3 ) ( 3 ) 9 =? 3 8 7 ( 3 ) 3 + 3 =? 8 3 3 = 3 3 =? 9 8( 3) ( 3) 9 ( 3) 3 + 3 =? 7 0 0 Division b zero is undefined. Intersection X=. Y=- 0 The pprent solution = 3 is etrneous. So, the onl solution is = 3. Monitoring Progress Solve the eqution. Check our solution(s). Help in English nd Spnish t BigIdesMth.com 7. 9 + + = 9 8. 7 = 39 Chpter 7 Rtionl Functions
Using Inverses of Functions Finding the Inverse of Rtionl Function Consider the function f() =. Determine whether the inverse of f is function. + 3 Then find the inverse. Grph the function f. Notice tht no horizontl line intersects the grph more thn once. So, the inverse of f is function. Find the inverse. Check f g 7 = + 3 = + 3 Set equl to f(). Switch nd. ( + 3) = Cross multipl. + 3 = Divide ech side b. f() = + 3 7 = 3 Subtrct 3 from ech side. So, the inverse of f is g() = 3. REMEMBER In prt (b), the vribles re meningful. Switching them to find the inverse would crete confusion. So, solve for m without switching vribles. Solving Rel-Life Problem 0m + 000 In Section 7. Emple, ou wrote the function c =, which represents m the verge cost c (in dollrs) of mking m models using 3-D printer. Find how mn models must be printed for the verge cost per model to fll to $90 b () solving n eqution, nd (b) using the inverse of the function.. Substitute 90 for c nd solve b cross multipling. 0m + 000 90 = m 90m = 0m + 000 0m = 000 m = b. Solve the eqution for m. 0m + 000 c = m c = 0 + 000 m c 0 = 000 m m = 000 c 0 000 When c = 90, m = 90 0 =. So, the verge cost flls to $90 per model fter models re printed. Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. Consider the function f() =. Determine whether the inverse of f is function. Then find the inverse. 0m + 800 0. WHAT IF? How do the nswers in Emple chnge when c =? m Section 7. Solving Rtionl Equtions 39
7. Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check. WRITING When cn ou solve rtionl eqution b cross multipling? Eplin.. WRITING A student solves the eqution 3 = nd obtins the solutions 3 nd. Are either 3 of these etrneous solutions? Eplin. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 0, solve the eqution b cross multipling. Check our solution(s). (See Emple.) 3. = +. = 9 + 7. 9. + 7 = 3 + = 3.. 8. 0. 9 3 = + 8 3 = = 8 + 3 = 7. USING EQUATIONS So fr in our vollebll prctice, ou hve put into pl 37 of the serves ou hve ttempted. Solve the eqution 90 00 = 37 + to find + the number of consecutive serves ou need to put into pl in order to rise our serve percentge to 90%.. MODELING WITH MATHEMATICS You hve 0. liter of n cid solution whose cid concentrtion is moles per liter. You wnt to dilute the solution with wter so tht its cid concentrtion is onl moles per liter. Use the given model to determine how mn liters of wter ou should dd to the solution. Concentrtion of new solution = Concentrtion of originl solution Volume of originl solution + Volume of originl solution Volume of wter dded USING STRUCTURE In Eercises 8, identif the LCD of the rtionl epressions in the eqution.. 7. + 3 + = 3 + + + =. 8. 7 = 9 + 9 + 3 = 0 3 In Eercises 9 30, solve the eqution b using the LCD. Check our solution(s). (See Emples 3 nd.). USING EQUATIONS So fr this bsebll seson, ou hve hits out of 0 times t-bt. Solve the eqution 0.30 = + to find the number of consecutive hits 0 + ou need to rise our btting verge to 0.30. 3. MODELING WITH MATHEMATICS Brss is n llo composed of % copper nd % zinc b weight. You hve ounces of copper. How mn ounces of zinc do ou need to mke brss? (See Emple.) 39 Chpter 7 Rtionl Functions 9.. 3.. 7. 9. 30. 3 + = 0. 3 + = 3 3 + = 3 + + = + 8 3 3 = + + + = + + = + 3 + = + 3... 8. 3 + = 3 0 + 3 = + 9 0 + = + 3 3 + = +
ERROR ANALYSIS In Eercises 3 nd 3, describe nd correct the error in the first step of solving the eqution. 3. 3. 3 + = 3 3 3 + 33 = 7 + + + = 0 3 3 ( + )3 7 + + + = 0 3 3 ( + )3 33. PROBLEM SOLVING You cn pint room in 8 hours. Working together, ou nd our friend cn pint the room in just hours.. Let t be the time (in hours) our friend would tke to pint the room when working lone. Cop nd complete the tble. (Hint: (Work done) = (Work rte) (Time)) Work rte Time Work done You room 8 hours hours Friend hours b. Eplin wht the sum of the epressions represents in the lst column. Write nd solve n eqution to find how long our friend would tke to pint the room when working lone. 3. PROBLEM SOLVING You cn clen prk in hours. Working together, ou nd our friend cn clen the prk in just. hours.. Let t be the time (in hours) our friend would tke to clen the prk when working lone. Cop nd complete the tble. (Hint: (Work done) = (Work rte) (Time)) Work rte Time Work done You prk hours. hours Friend. hours b. Eplin wht the sum of the epressions represents in the lst column. Write nd solve n eqution to find how long our friend would tke to clen the prk when working lone. 3. OPEN-ENDED Give n emple of rtionl eqution tht ou would solve using cross multipliction nd one tht ou would solve using the LCD. Eplin our resoning. 3. OPEN-ENDED Describe rel-life sitution tht cn be modeled b rtionl eqution. Justif our nswer. In Eercises 37, determine whether the inverse of f is function. Then find the inverse. (See Emple.) 37. f() = 7 38. f() = + 39. f() = 3 0. f() =. f() = 8. f() = 9 + 3. f() = +. f() = 7. PROBLEM SOLVING The cost of fueling our cr for er cn be clculted using this eqution: Fuel cost for er = Miles driven Fuel-efficienc rte Price per gllon of fuel Lst er ou drove 9000 miles, pid $3. per gllon of gsoline, nd spent totl of $389 on gsoline. Find the fuel-efficienc rte of our cr b () solving n eqution, nd (b) using the inverse of the function. (See Emple.). PROBLEM SOLVING The recommended percent p (in deciml form) of nitrogen (b volume) in the ir tht diver brethes is given b p = 0.07, where d is the d + 33 depth (in feet) of the diver. Find the depth when the ir contins 7% recommended nitrogen b () solving n eqution, nd (b) using the inverse of the function. Section 7. Solving Rtionl Equtions 397
USING TOOLS In Eercises 7 0, use grphing clcultor to solve the eqution f() = g(). 7. f() = 3, g() = 8. f() = 3, g() = 9. f() = +, g() = 0. f() = +, g() = +. MATHEMATICAL CONNECTIONS Golden rectngles re rectngles for which the rtio of the width w to the length is equl to the rtio of to + w. The rtio of the length to the width for these rectngles is clled the golden rtio. Find the w vlue of the golden rtio using rectngle with width of unit.. HOW DO YOU SEE IT? Use the grph to identif the ( ) solution(s) of the rtionl eqution = +. Eplin our resoning. = ( ) = + USING STRUCTURE In Eercises 3 nd, find the inverse of the function. (Hint: Tr rewriting the function b using either inspection or long division.) 3. f() = 3 +. f() = 7 + 3. ABSTRACT REASONING Find the inverse of rtionl functions of the form = + b. Verif our nswer c + d is correct b using it to find the inverses in Eercises 3 nd.. THOUGHT PROVOKING Is it possible to write rtionl eqution tht hs the following number of solutions? Justif our nswers.. no solution b. ectl one solution c. ectl two solutions d. infinitel mn solutions 7. CRITICAL THINKING Let be nonzero rel number. Tell whether ech sttement is lws true, sometimes true, or never true. Eplin our resoning.. For the eqution =, = is n etrneous solution. 3 b. The eqution = hs ectl one solution. c. The eqution = + + hs no solution. 8. MAKING AN ARGUMENT Your friend clims tht it is not possible for rtionl eqution of the form = c, where b 0 nd d 0, to hve b d etrneous solutions. Is our friend correct? Eplin our resoning. Mintining Mthemticl Proficienc Is the domin discrete or continuous? Eplin. Grph the function using its domin. (Skills Review Hndbook) 9. The liner function = 0. represents the mount of mone (in dollrs) of qurters in our pocket. You hve mimum of eight qurters in our pocket. 0. A store sells broccoli for $ per pound. The totl cost t of the broccoli is function of the number of pounds p ou bu. Evlute the function for the given vlue of. (Section.) Reviewing wht ou lerned in previous grdes nd lessons. f() = 3 + 7; =. g() = + 7 3 + ; = 3 3. h() = 3 + 3 + ; = 3. k() = 3 + ; = 398 Chpter 7 Rtionl Functions
7.3 7. Wht Did You Lern? Core Vocbulr rtionl epression, p. 37 simplified form of rtionl epression, p. 37 comple frction, p. 387 cross multipling, p. 39 Core Concepts Section 7.3 Simplifing Rtionl Epressions, p. 37 Multipling Rtionl Epressions, p. 377 Dividing Rtionl Epressions, p. 378 Section 7. Adding or Subtrcting with Like Denomintors, p. 38 Adding or Subtrcting with Unlike Denomintors, p. 38 Simplifing Comple Frctions, p. 387 Section 7. Solving Rtionl Equtions b Cross Multipling, p. 39 Solving Rtionl Equtions b Using the Lest Common Denomintor, p. 393 Using Inverses of Functions, p. 39 Mthemticl Prctices. In Eercise 37 on pge 38, wht tpe of eqution did ou epect to get s our solution? Eplin wh this tpe of eqution is pproprite in the contet of this sitution.. Write simpler problem tht is similr to Eercise on pge 38. Describe how to use the simpler problem to gin insight into the solution of the more complicted problem in Eercise. 3. In Eercise 7 on pge 390, wht conjecture did ou mke bout the vlue the given epressions were pproching? Wht logicl progression led ou to determine whether our conjecture ws correct?. Compre the methods for solving Eercise on pge 397. Be sure to discuss ss the similrities nd differences between the methods s precisel s possible. Performnce Tsk Circuit Design A thermistor is resistor whose resistnce vries with temperture. Thermistors re n engineer s drem becuse the re inepensive, smll, rugged, nd ccurte. The one problem with thermistors is their responses to temperture re not liner. How would ou design circuit tht corrects this problem? To eplore the nswers to these questions nd more, go to BigIdesMth.com. 399