Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.


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1 Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo with questions.
2 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse function for ll 6= 0. Find f lgericll nd show tht f is selfinverse function. 8 Show tht f : 7! 8, 6= is selfinverse function : reference to its grph using lger. 9 Consider the function f() =. Find f (). Find: i (f ± f )() ii (f ± f)(). 0 Consider the functions f : 7! + nd g : 7! 8. Find g ( ). Solve for if (f ± g )() =9. Consider the functions f : 7! nd g : 7! p. Find: i f() ii g (): Solve the eqution (g ± f)() =. Given f : 7! nd g : 7!, show tht (f ± g )() =(g ± f) (). Which of these functions is selfinverse function, so f () =f()? f() = f() = c f() = d f() = e f() = 6 The horizontl line test ss: For function to hve n inverse function, no horizontl line cn cut its grph more thn once. Eplin wh this is vlid test for the eistence of n inverse function. Which of the following functions hve n inverse function? i ii iii  REVIEW SET A NONCALCULATOR If f() = find: f() f( ) c f( ) If f() = + where nd re constnts, find nd for f() = 7 nd f() =. If g() =, find in simplest form: g( + ) g( )
3 66 FUNCTIONS (Chpter ) For ech of the following grphs determine: i the rnge nd domin ii the nd intercepts iii whether it is function. \Wl_T_ (, ) Drw sign digrm for: ( + )( ) If f() = +, f() = nd f () =, find nd. 7 Cop the following grphs nd drw the inverse function on the sme set of es: 8 Find f () given tht f() is: + 9 Given f : 7! +6 nd h : 7!, show tht (f ± h )() =(h ± f) (). REVIEW SET B For ech of the following grphs, find the domin nd rnge: ( )( ) (, ) CALCULATOR If f() = nd g() = +, find: (f ± g)() (g ± f)() Drw sign digrm for:
4 FUNCTIONS (Chpter ) 67 Consider f() =. c For wht vlue of is f() meningless? Sketch the grph of this function using technolog. Stte the domin nd rnge of the function. + Consider the function f() =. Find nd given tht = f() hs smptotes with equtions = nd =. Write down the domin nd rnge of f (). + 6 Consider the function f : 7!. Determine the equtions of the smptotes. c d Discuss the ehviour of the function s it pproches its smptotes. Determine the es intercepts. Sketch the grph. 7 Consider the functions f() = + nd g() =. Find (g ± f)(). Given (g ± f)() =, solve for. c Let h() =(g ± f)(), 6=. i Write down the equtions of the smptotes for the grph of h(). ii Sketch the grph of h() for 6 6. iii Stte the domin nd rnge of h(). 8 Consider f : 7! 7. On the sme set of es grph =, f nd f. c Find f () using vrile interchnge. Show tht f ± f = f ± f =, the identit function. 9 The grph of the function f () =, is shown longside. Sketch the grph of = f (). Stte the rnge of f. c Solve: i f () = 0 ii f () = (, ) (, )
5 68 FUNCTIONS (Chpter ) REVIEW SET C For ech of the following grphs, find the domin nd rnge: ( ) ( ) ( ) If h() =7 : find in simplest form h( ) find if h( ) =. If f() = nd g() = p, find in simplest form: (f ± g)() (g ± f)() Find, nd c if f(0) =, f( ) = nd f() = nd f() = ++c. Cop the following grphs nd drw the grph of ech inverse function on the sme set of es: 6 For ech of the following functions f() find f () : f() =7 f() = + 7 Given f : 7! nd h : 7!, show tht (f ± h )() =(h ± f) (). 8 Given f() = + nd g() =, find (g ± f )().
6 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) For the grph of = f() given, sketch the grph of: =f() = f() c = f( + ) d = f() e = f( ) ( ) g() For the grph of = g() given, sketch the grph of: = g()+ = g() c = g( ) d = g( + ) 6 For the grph of = h() given, sketch the grph of: = h()+ = h() c = h( ) d = h h() ( ) REVIEW SET A NONCALCULATOR If f() =, find in simplest form: f() f( ) c f() d f( ) e f() If f() =, find in simplest form: ³ f() f( ) c f( ) d f e f() f( ) Consider f : 7!. Sketch the function f. Find lgericll the i intercept ii intercept iii grdient of the line. c i Find when =0:: ii Find when =0:7: The grph of f() = + + is trnslted to its imge g() the vector. Write the eqution of g() in the form g() = + +c+d. ³
7 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) The grph of = f() is shown longside. The is is tngent to f() t = nd f() cuts the is t =. On the sme digrm sketch the grph of = f( c) where 0 <c<. Indicte the coordintes of the points of intersection of with the is. z ( ) 6 For the grph of = f(), sketch grphs of: = f( ) = f() c = f( + ) d = f()+ ( ) ( ) ( ) ( ) 7 The grph of = f() is shown longside. Sketch the grph of = g() where g() =f( + ). Stte the eqution of the verticl smptote of = g(). c Identif the point A 0 on the grph of = g() which corresponds to point A. ( ) A (, ) 8 Consider the function f : 7!. On the sme set of es grph: = f() = f( ) c =f( ) d =f( ) + REVIEW SET B If f() =, find in simplest form: f( ) f( + ) c f()+ Consider f() =( + ). Use our clcultor to help grph the function. Find: i the intercepts ii the intercept. c Wht re the coordintes of the verte of the function? CALCULATOR Consider the function f : 7!. On the sme set of es grph: = f() = f( + ) c =f( + ) d =f( + )
8 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) Consider f : 7!. Use our clcultor to help grph the function. True or flse? i As!,! 0: ii As!,! 0: iii The intercept is : iv > 0 for ll. The grph of the function f() =( + ) + is trnslted units to the right nd units up. c Find the function g() corresponding to the trnslted grph. Stte the rnge of f(). Stte the rnge of g(). 6 For ech of the following functions: i ii iii iv Find = f(), the result of trnsltion ³. Sketch the originl function nd its trnslted function on the sme set of es. Clerl stte n smptotes of ech function. Stte the domin nd rnge of ech function. = = c = log 7 Consider the function g() =( + ). Use technolog to help sketch grph of the function. Find the es intercepts. c Find the coordintes of the verte of g(). 8 Sketch the grph of f() = +, nd on the sme set of es sketch the grphs of: f() f() c f()+ REVIEW SET C If f() =, find in simplest form: ³ f( ) f() c f d f( + ) Sketch the grph of f() =, nd on the sme set of es sketch the grph of: = f( ) = f() c = f() d = f( )
9 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) The grph of cuic function = f() is shown longside. ( ) Sketch the grph of g() = f( ). Stte the coordintes of the turning points of = g(). (, ) (, ) The grph of f() = is trnsformed to the grph of g() reflection nd trnsltion s illustrted in the digrm longside. Find the formul for g() in the form g() = + + c. V (, ) ( ) g() Given the grph of = f(), sketch grphs of: f( ) f( + ) c f(). ( ) (' Ow_\) 6 The grph of f() = + + is trnslted to its imge, = g(), the vector. Write the eqution of g() in the form g() = + + c + d. 7 Find the eqution of the line tht results when the line f() = + is trnslted: i units to the left ii 6 units upwrds. Show tht when the liner function f() =+ is trnslted k units to the left, the resulting line is the sme s when f() is trnslted k units upwrds.
10 Grphing Rtionl Functions Worksheet  All grphs must e done on grph pper. No Clcultor: For ech of the following rtionl functions ou should ) find n horizontl smptotes, verticl smptotes or olique smptotes; ) find n intercept(s) nd the intercept; c) find the coordintes of n hole(s) in the grph; d) write down the end ehvior of the function; e) grph the functions without clcultor.. f ( ) = 9. g( ) =. ( + ) h:. + i: + Clcultor Allowed: For ech of the following rtionl functions ou should ) find n horizontl smptotes, verticl smptotes or olique smptotes; ) find n intercept(s) nd the intercept; c) find the coordintes of n hole(s) in the grph; d) write down the end ehvior of the function; e) grph the functions with the id of our clcultor.. f ( ) = 6. g : + 7. h( ) + + = + 8. i( ) = + 8 8
11 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 8 0 The totl cost of producing tosters per d is given C = euros, nd the selling price of ech toster is ( ) euros. How mn tosters should e produced ech d in order to mimise the totl profit? A mnufcturer of reques knows tht if of them re mde ech week then ech one will cost ( ) pounds nd the totl receipts per week will e (000 ) pounds. How mn reques should e mde per week to mimise profits? INVESTIGATION Answer the following questions: + + c =0 hs roots p nd q. Prove tht p + q = nd pq = c. SUM AND PRODUCT OF ROOTS +=0 hs roots p nd q. Without finding the vlues of p nd q, find: p + q pq c p + q d p + q Find ll qudrtic equtions with roots which re: one more thn the roots of +=0 the squres of the roots of +=0 c the reciprocls of the roots of +=0. REVIEW SET 6A NONCALCULATOR Consider the qudrtic function = ( + )( ). Stte the intercepts. Stte the eqution of the is of smmetr. c Find the intercept. d Find the coordintes of the verte. e Sketch the grph of the function. Solve the following equtions, giving ect nswers: =0 0 = 0 c = 60 Solve using the qudrtic formul: + +=0 + =0 Solve the following eqution completing the squre : +7 =0 Use the verte, is of smmetr nd intercept to grph: =( ) = ( + ) +6 6 Find, in the form = + + c, the eqution of the qudrtic whose grph: touches the is t nd psses through (, ). hs verte (, ) nd psses through (, ).
12 86 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 7 Find the mimum or minimum vlue of the reltion = + + nd the vlue of for which the mimum or minimum occurs. 8 Find the points of intersection of = nd =. 9 For wht vlues of k does the grph of = + + k not cut the is? 0 Find the vlues of m for which + m =0 hs: repeted root two distinct rel roots c no rel roots. The sum of numer nd its reciprocl is 0. Find the numer. Show tht no line with intercept of (0, 0) will ever e tngentil to the curve with eqution = +7. The digrm shows qudrtic f() = + m + n. (, ) Determine the vlues of m nd n. Find k given tht the grph psses through the point (, k). c Stte the verte of = g() given g() =f( ) +. d Find the domin nd rnge of f() nd g(). REVIEW SET 6B CALCULATOR Consider the qudrtic function = +6. Convert it into the form = ( h) + k completing the squre. Stte the coordintes of the verte. c Find the intercept. d Sketch the grph of the function. e Use technolog to check our nswers. Use technolog to solve: ( )( + ) = = Drw the grph of = +. Find the eqution of the is of smmetr nd the verte of = Using the discriminnt onl, determine the nture of the solutions of: 7=0 + 8 = 0 6 If [AB] hs the sme length s [CD], [BC] is cm shorter thn [AB], nd [BE] is 7 cm in length, find the length of [AB]. A E B D C
13 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 87 7 For wht vlues of c do the lines with equtions = + c intersect the prol = + in two distinct points? Choose one such vlue of c from prt nd find the points of intersection. 8 For the qudrtic = +, find: the eqution of the is of smmetr the coordintes of the verte c the es intercepts. d Hence sketch the grph. 9 An open squre continer is mde cutting cm squre pieces out of piece of tinplte. If the cpcit is 0 cm, find the size of the originl piece of tinplte. 0 Find the points where = + nd = + + meet. Find the mimum or minimum vlue of the following qudrtics, nd the corresponding vlue of : = + +7 = m of fencing is used to construct m 6 rectngulr niml pens s shown. m Show tht =. 9 Find the re A of ech pen in terms of. c Find the dimensions of ech pen if ech pen is to hve mimum re. d Wht is the mimum re of ech pen? Two different qudrtic functions of the form f() =9 k + ech touch the is. Find the two vlues of k. Find the point of intersection of the two qudrtic functions. c Descrie the trnsformtion which mps one function onto the other. REVIEW SET 6C Consider the qudrtic function = ( ). Stte the eqution of the is of smmetr. Find the coordintes of the verte. c Find the intercept. d Sketch the grph of the function. e Use technolog to check our nswers. Solve the following equtions: =0 7 =0 Solve the following using the qudrtic formul: 7 +=0 +=0
14 88 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) Find the eqution of the qudrtic reltion with grph: c Use the discriminnt onl to find the reltionship etween the grph nd the is for: = + 7 = Determine if the qudrtic functions re positive definite, negtive definite or neither: = + + = Find the eqution of the qudrtic reltion with grph: (, ) 8 In right ngled tringle, one leg is 7 cm longer thn the other, nd the hpotenuse is cm longer thn the longer leg. Find the length of the hpotenuse. 9 Find the intercept of the line with grdient tht is tngentil to the prol = +. 0 For wht vlues of k would the grph of = + k cut the is twice? Find n epression for the qudrtic which cuts the is t nd nd hs intercept. Give our nswer in the form = + + c. For wht vlues of m re the lines = m 0 tngents to the prol = +7 +? The digrm shows prol = ( + m)( + n) where m>n. A B Find, in terms of m nd n, the: i coordintes of the intercepts A nd B ii eqution of the is of smmetr. Stte the sign of: i the discriminnt ii.
15 ANSWERS 697 EXERCISE H i ii, iii f () = i f () = ii c i f () = c e \Tw_ \Qe_ f j 6 6 0g f j g c f j g d f j 6 6 0g f j 6 <g \Tw_ \Qe_ ii i ii, iii f () = i f () = + ii d f 6 Er_ Ew_ Er_ Qw_ Ew_ (, ) Qw_ 7 f : 7!, 6= 0 stisfies oth the verticl nd horizontl line tests nd so hs n inverse function. f () = nd f() = ) f = f ) f is selfinverse function 8 = 8 is smmetricl out =, ) f is selfinverse function. f () = 8 REVIEW SET A 0 c = 6, = i Rnge = f j > g, Domin = f j R g ii int,, int iii is function 9 i Rnge = f j =or g, Domin = f j R g ii no intercepts, intercept iii is function \We_  6 =, = 7 8 f () = f () = 9 (f ± h )() =(h ± f) () = REVIEW SET B Domin = f j R g, Rnge = f j > g Domin = f j 6= 0, g, Rnge = f j 6 or >0g + + =0 nd f() = 8 ) f = f ) f is selfinverse function 9 f () = + i (f ± f )() = ii (f ± f)() = 0 0 = i ii 6 = (f ± g )() = + nd (g ± f) () = Is not Is c Is d Is e Is i is the onl one c Domin = f j 6= 0g, Rnge = f j >0g
16 698 ANSWERS =, = Domin = f j 6= g, Rnge = f j 6= g 6 verticl smptote =, horizontl smptote = s!,! s!,! s! +,! s!,! + c intercept, intercept d 7 (g ± f)() = = + c i verticl smptote =, horizontl smptote =0 ii \Qe_ iii Domin = f j 6= g, Rnge = f j 6= 0g 8 f () = +7 \Qw_ 9 Rnge = f j g c i ¼ :8 ii = REVIEW SET C \Qw_ 0 6 = p p =, = 6, c = () (, ) (, ) Qw_ \Qr_ ( ) (, ) f ( ) (, ) f ( ) Domin = f j > g, Rnge = f j 6 <g Domin = f R g, Rnge = f j =, or g 6 f () = 7 f () = 7 (f ± h )() =(h ± f) +6 () = 8 6 EXERCISE A,,,, 9,, 7 c, 6, 8, d 96, 8,, Strts t 8 nd ech term is 8 more thn the previous term. Net two terms 0, 8. Strts t, ech term is more thn the previous term;, 7. c Strts t 6, ech term is less thn the previous term; 6,. d Strts t 96, ech term is 7 less thn the previous term; 68, 6. e Strts t, ech term is times the previous term; 6, 0. f Strts t, ech term is times the previous term; 6, 86. g Strts t 80, ech term is hlf the previous term; 0,. h Strts t, ech term is of the previous term;,. i Strts t 0 000, ech term is of the previous term; 80, 6. Ech term is the squre of the term numer;, 6, 9. Ech term is the cue of the term numer;, 6,. c Ech term is n(n + ) where n is the term numer; 0,, 6. 79, 7 80, 0 c 6, 96 d, 7 e 6, f, 8 EXERCISE B,, 6, 8, 0, 6, 8, 0, c,,, 7, 9 d,,,, 7 e, 7, 9,, f,, 7, 9, g, 7, 0,, 6 h,, 9,, 7,, 8, 6, 6,,, 8, 96 c, d,, 8, 6,,, 8, 6 7,,,, 7 EXERCISE C 7 6 c 0 07 c + d u =6, d = u n = n c d es, u 0 e no u = 87, d =, u n = 9 n c 69 d u 97 u =, d = c 69 d u = 6 u =, d = 7 c 7 d n > 68 7 k = 7 k = c k = d k =0 e k = or f k = or
17 ANSWERS 709 iii g( ) g( ) g( ) g( ) g( ) = f( ) is the reflection of = f() in the is. i (, 0) ii (, ) iii (, ) i (7, ) ii (, 0) iii (, ) 6 i (, ) ii (0, ) iii (, ) i (, ) ii (0, ) iii (, ) 7 A rottion out the origin through 80 o. (, 7) c (, ) 6 h( ) h( ) h( ) h( ) h( ) EXERCISE B. ( ) ( ) ( ) ( ) REVIEW SET A 8 c d + e 6 c + + d e + i ii iii c i = : ii =0:9 c ( ) ( ) g() = + 6 ( ) ( ) ( ) c ( ) ( ) ( ) 6 c c c ( ) ( c) f ( ) f ( ) f ( ) f ( ) f ( ) A B c D d C 7 g() ( ) f ( ) f ( ) f () f ( ) f ( ) f ( ) A' (, ) = c A 0 (, ) A (, )
18 70 ANSWERS 8 i = ii REVIEW SET B c V i nd ii c V(, ) i true iii flse ii flse iv true 6 i = ii c f ( ) f ( ) f ( ) f ( ) iii For =, VA is =0, HA is =0 For =, VA is =, HA is = iv For =, domin is f j 6= 0g, rnge is f j 6= 0g For =, domin is f j 6= g, rnge is f j 6= g f ( ) f ( ) f ( ) f ( ) g() =( ) +8 f j > g f j > 8g iii For =, HA is =0, no VA For =, HA is =, no VA iv For =, domin is f j R g, rnge is f j >0g For =, domin is f j R g, rnge is f j > g c i = log ( ) ii iii For = log, VA is =0, no HA For = log ( ), VA is =, no HA iv For = log, domin is f j >0g, rnge is f R g For = log ( ), domin is f j >g, rnge is f R g 7 intercepts nd, intercept c (, ) 8 REVIEW SET C log ( ) g ( ) ( )X c 8 log f ( ) f ( ) f () f ( ) d 0 + f ( ) f ( ) f ( ) f () f ( )
19 ANSWERS 7 (, ) nd ( ) (, 0) g() = g() = i = +8 ii = +8 f( + k) =( + k)+ = + + k = f()+k EXERCISE 6A. =0, 7 =0, c =0, 7 d =0, e =0, 8 f =0, g =, h =, i =, 7 j = k =, l =, = =, 7 c =, 6 d =, e =, f =, g =, h =, i =, j =, k = 8, l =, 7 =, =, c =0, d =, e =, f = EXERCISE 6A. = p no rel solns. c = p d =8 p 7 e = p f = p 6 g = p 0 h = p p i = 7 = p = p 7 c =7 p d = p 7 e = p f = p 7 g = p h = p 6 i no rel solns. = p = p 9 c = p 7 p d = 7 p e = 7 f = p 0 6 EXERCISE 6A. (, ) (, ) (, ) (, ) = p 7 = p c = p d = p e = p f = p 7 g = p 9 7 h = 7 p 9 97 = p = p 8 7 c = p 8 d = p p 7 e = f = p 7 g() ('\\Ow_) f ( ) f ( ) f ( ) f ( ) EXERCISE 6B rel distinct roots rel distinct roots c rel distinct roots d no rel roots e repeted root, c, d, f = 6 m i m = ii m< iii m> =9 8m i m = 9 8 ii m< 9 8 iii m> 9 8 c =9 m i m = 9 ii m< 9 iii m> 9 =k +8k i k< 9 iii k = 9 f = k k EXERCISE 6C. or k> ii k 6 9 or k > or iv 9 <k< i <k<0 ii 6 k 6 0 iii k = or 0 iv k< or k>0 =( )( + ) = ( )( + ) c = ( + )( + ) d = ( + ) 0 8 Oi_ Or_ 0 i k< 8 or k>0 ii k 6 8 or k > 0 iii k = 8 or 0 iv 8 <k<0 = k i <k< ii 6 k 6 iii k = iv k< or k> c =k +k i k< 6 or k> ii k 6 6 or k > iii k = 6 or iv 6 <k< d =k k 6 i k< or k>6 ii k 6 or k > 6 iii k =6or iv <k<6 e =9k k 9 \Ql_E_ \Re_ 0 m m m k k k k k k
20 . f ( ) = 9 HA: = 0 VA: =, =  OA: none Hole: none int(s): (0,0) int: (0,0) End ehvior: s, 0 s, 0. g( ) = HA: none VA: = OA: = Hole: none int(s): (,0),(,0) int: (0,) End ehvior: s, s,
21 . h: ( + ) + HA: = 0 VA: = OA: none Hole:, 7 int(s): none int: 0, End ehvior: s, 0 s, 0. i: + HA: = VA: = OA: none Hole: none int(s): int: 0,,0 End ehvior: s, s,
22 . f ( ) = HA: = VA: =, =  OA: none Hole: none int(s): (0,0) int: (0,0) End ehvior: s, s, 6. g : + HA: none VA: = OA: =  + Hole: none int(s): none int: (0,.) End ehvior: s, s,
23 7. h( ) + + = + HA: none VA: none OA: = + Hole: none int(s): (.9,0) int: (0,) End ehvior: s, s, 8. i( ) = HA: none VA: = OA: none Hole:, 6 int(s): (0,0), (,0) int: (0,0) End ehvior: s, s, Fun Fct: This grph ctull hs prolic smptote, ut ou do not hve to know how to find it.
24 7 ANSWERS EXERCISE 6G min., when = m. 8, when = c m. 8, when = d min. 8, when = e min. 6 when = 8 f m. 6 8, when = 7 0 refrigertors, $ m 0 m c 00 m : m 6 m m 0 m m 7 8 units 8 =6 cm cm REVIEW SET 6A, e = c d (, 9 ) =0or = = or or c = p = p 6 = 7 p 6 = + 8 = = which is < 0 ) m. m. =when = 8 (, ) nd (, 8) 9 k< 8 0 m = 9 8 m< 9 8 c m> or 6 m =, n = k =7 c (, ) d f() hs domin f j R g, rnge f j > g g() hs domin f j R g, rnge f j > g REVIEW SET 6B (') = + d (, ) c ¼ 0:86 or : ¼ 0:86 or :686 V( ) =, V(, ) two distinct rtionl roots repeted root  &\Qw_\'  &\Ew_\'\Qs_T_*  6 :9 cm 7 c> 6 emple: c =, (, ) nd (, 7) 8 = d (, ) c intercept, ints. p 6 9 : cm : cm 0 touch t (, 9) min. = when = m. = 8 when = ³ A = c 7 9 m m k = or (0, ) c horizontl trnsltion of units REVIEW SET 6C = d (, ) c = p 7 = 7 p 7 = 7 p 7 no rel roots = 0 9 ( ) 0 = ( )( 7) 7 c = ( + ) 9 grph cuts is twice grph cuts is twice 6 neither positive definite d 0 m 7 = ( )( + ) = 6( ) cm 9 0 k< = + + m = or 9 i A( m, 0), B( n, 0) m n ii = i positive ii negtive EXERCISE p +p q +pq + q c d e f g h p p q +6p q pq + q c d e f g h \Qw_\~`6 +\Qw_\~`6  (') (')
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