NAME: M. WAIN FUNCTIONS
evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors where necessr Fctorise Let Hnd Side Use Null Fctor Lw NFL Fctorising Qudrtic b c o I b c the discriminnt, is: Perect squre, use criss-cross Not perect squre, but positive use the qudrtic ormul, b b c Negtive, NO EAL solution Cubic b c d o Use grouping i possible o Fctor theorem nd long division Qurtic b c d e o Use substitution, eg let o Fctor theorem < 0, no solutions = 0, solution >0, solutions
Emples: Solve the ollowing or : 0 0 0 0 7 or,, 0 0 0 0 0 division Long P o ctor is P P Let 0 0 0 or or Let 0 or
0 Let 0 0 0 0 0 0 0 or 0 iii Itertion Use itertion to ind the solutions to correct to deciml point. LetP P0 P P P P thereore solutions close to & tr P.7 0. P. 0.0. is solution to d. p. P tr P.7 0. P. 0.0. is solution to d. p. Questions: question sheet. see Below Check using solve commnd on Ti-NSpire
Solving Polnomil Equtions 0 Question Sheet one term in > term in Solve the ollowing: Question Hint. -. Eliminte the denomintor 0. 9 -. 7 00 Collect like terms, nswer to d.p. Eliminte denomintors in one go. - 0 7. -. 0 i ect nswer ii correct to d.p. 9. Correct to d.p. 0. 0 -. Mke HS=0, then ctor theorem. 0 Grouping is quicker. 0 Let =. Mke HS=0, let =, ns. to d.p.. 7 0 Grouping two nd three. Let =+ 7. Eliminte denomintor then ctor 0 theorem. Eliminte denomintor then let = 9. 0 Eliminte constnt, HCF 0. Eliminte constnts For the ollowing use itertion to ind solutions correct to deciml plce d.p. 7 0. 0
Completing The Squre Completing the squre llows qudrtic o the orm n m, to be written in the Turning Point orm, c b. Emples: Epress the ollowing in the Turning Point orm c b. 9 9 7 7 7 E A Q, Onl Complete the Squre CAS Clcultor: epnd commnd to check nswers & the Complete The Squre commnd
Fctor Theorem Emples Without perorming long division, ind the reminder when is divided b. -9. Sothe reminder is 9 P P Let Find " ", given tht when is divided b the reminder is 7. P P Let 7 7 7 tionl oot Theorem Sometimes there re no integer solutions to polnomil, but there mbe rtionl solutions. e.g. i P, we cn show P 0, P- 0, P 0, P- 0. So there is no integer solution. So net we tr P, P, P & P nd will discover 0 P thereore is ctor o P. Unsure o signs then solve the eqution = 0 or. Emple: Use the tionl-root theorem to help ctorise P,, be would : The solutions the squre b completing. 0 90 0 or Note ctor is So P P P P P P ED Q,, 7,, 0,,, 7, 0
Stright Lines/Simultneous Equtions The grdient o stright line is lws constnt. Grdient m rise run Distnce between points: d Pthgors "m" is negtive "m" is positive "m" is zero "m" is undeined Midpoint, M, o two points is given b I m is the grdient o stright line nd m is the grdient o nother stright line o I the two lines re prllel then m m o I the two lines re perpendiculr then m m or m m Eqution o stright line: To ind the eqution o stright line, ou need: o The grdient m nd the Y-intercept c, then use m c o The grdient m nd the coordintes o one point on the line m c or m. o The coordintes o two points on the line,, nd then use m c or m. o Emple: Find the eqution o the line pssing through -, - nd,.,,, then use, then use m, nd Let,, nd,, m EC Q,,,, 0,,,,,,
Simultneous Equtions situtions o No solutions o Ininitel mn solutions o A unique solution. No solution Mens the lines re prllel The hve the sme grdient but dierent Y-intercept e.g. &. Ininitel mn solutions Mens ou hve the sme line e.g. &. A unique solution Mens the lines re dierent nd meet t one point onl. e.g. & Emple : Eplin wh the ollowing pir o simultneous equtions hve no solutions & Prllel lines, sme grdient, dierent Y - Intercept Emple : Consider the sstem o simultneous equtions given b: m m m Find the vlues o m or which there is no solution.
m m m m or no solutions or ininite solutions the determinnt o the mtri m tht is 0 m m m 0 m m m 0 m Check : m : : m - : m 0 or m equls zero m the re the sme eqution, hence n ininite number o solutions. : prllel lines sme grdient, dierent - int no solution this is the nswer required. Note: For unique solution the determinnt 0. For the bove emple: => the vlues o m or which there is unique solution, m\{-, } Simultneous Liner Equtions Worksheet m m 0... m m 0 m,. Consider the sstem o simultneous liner equtions given b m m 0 b m m m m Find the vlues o m or which there is unique solution.. Consider the sstem o simultneous liner equtions given b m 7 m b m 0.7m m Find the vlues o m or which there re ininitel mn solutions.. Consider the sstem o simultneous liner equtions given b m m b m m m Find the vlues o m or which there is no solution. Answers: m \{,} b m \{,} m = b m = m = b m =, E F,,,
Sets Nottion A set is collection o objects The objects re known s elements o I is n element o A, A A is not n element o A or o the set o oddnumbers I something is subset o A, or emple B, B A emple o subset I sets hve common elements, it is clled n intersection ie it the empt set., union, A B is the set o elements tht re either in A or B.. Bos in the Yer Methods clss is n A B. The set dierence o two sets A nd B is given b A\B = {: A, B}. Mens wht s in A but not in B. Emple I A {,,,7} nd B {,,,,7} i Find A B b A B c A \ B A B {,7} b A B {,,,,,,7} c A \ B {,} d B \ A {,,} d B \ A ii True or Flse A b A c B d{,} A e{,} B True Flse Flse Flse True Sets o Numbers N, the set o Nturl Numbers {,,,,..} is subset o... Z, the set o Integers {.,-, -, 0,,,.} is subset o Q, the set o tionl numbers, numbers which cn be epressed in the orm n m is subset o, the set o el numbers N Z Q Q, is the set o irrtionl numbers, eg,,, e Q Z N
Subsets o the el numbers Set Intervl Number Line {: < < b}, b b {: < b} [, b b {: < b}, b] b {: b} [, b] b {: > }, {: } [, {: < } -, {: } -, ] Emple: Complete: 0 Set Intervl Number Line A {: >} B [-,] C 0 b D -, ] E b F G {: < 0} H \ {0} Eercise A Q,,,,,, 7,, 9
eltions nd Functions Deinition o unction An reltion in which no two ordered pirs hve the sme irst element ie vlue. o The vlue is onl used once o {,,,,,,,} is unction o {-,0, -, -, -,, 0,-, 0,} is not unction A unction is reltion with one-to-one correspondence or mn-to-one correspondence. Eg o tht: or or g Functions re subset o reltions one-to-mn or mn-to-mn I reltion is represented grphicll, ppl verticl line test to decide whether it is unction or not o Cuts the grph once unction o Cuts the grph more thn once - not unction The irst elements o the ordered pir in unction mkes the set clled the DOMAIN. The second elements mke the set clled the ANGE. Some other terms used: Imge, pre-imge,, Nottion or description o Function :, is the nme o the unction use, g, h, : mens such tht is the domin be creul or restrictions o the domin is the possible vlues tht the domin cn mp onto it is not the ctul rnge represents the rule Emple: ewrite the ollowing using the unction nottion {, : h 7, } h :,0, h 7 Emple: For the unction with the rule g, evlute: i g ii g- iii g0 iv g v g+h vi g=9 i g ii g iii g0 0 v g h h vi 9 iv h h g -nd re the pre - imge o 9. h h EB Q, ce,,,,, 7,, 9cde, 0,, bc,,,,
One to One Functions VETICAL LINE TEST to see i we hve unction HOIZONTAL LINE TEST to see i we hve one-to-one unction Emples: o Prbol o Cubic o Eponentil Implied domins Oten the domin is not stted or unction. Assume the domin is to be s lrge s possible i.e. select rom Emples: o o, the implied domin is s ll vlues o cn be used, the implied domin is [0, o, [, o,, ] o, \ {0} o, \{} o 7,,] [, look t grph o prbol wht is under the root. o, \{, 7 } E C Q,,,,,, 7,
Hbrid unctions Piecewise unctions A unction which hs dierent rules or dierent subsets o the domin. Emple: Sketch the grph o:, 0,0, nd stte the domin nd rnge. Domin =, 00,, nge = 0, or For the bove unction ind: b c b c i ii iii,,,, 0 0 0 0, 0,0, Ti-NSpire : From the templtes, select the hbrid with choices. see p o tet.
Odd & Even Functions An odd unction is deined b: odd is it i ODD/EVEN the TEST to seei is this Cn lso consider n Odd unction hs 0 o rottionl smmetr bout the origin. An even unction is deined b: even is it i E C Q 9, 0,,,,,,, 7,, 9
Sums nd Products o Functions Emple: I nd g ind: g b g c g d g onl deined domin or c g Implied domin or g, nd the implied domin or g, g, b g g g, dom d g Grphing b Additions o Ordintes This involves the ddition o the -vlues o the given equtions. For emple, i nd g the grph o g is obtined b dding the -vlues or ever vlue o or which both curves simultneousl eist. g, For the domin is [0, For g the domin is -, Thereore, the vlues o or which both curves re deined simultneousl is given b [0, Sketch the two grphs bove, on grph pper, see blckbord or speciic instructions. Adding the -vlues is stright orwrd s long s ou know the equtions o the grphs. However, ou need to be ble to dd two grphs without this inormtion. Hints: when using the ddition o ordintes.. Look or regions where both grphs re positive ie both lie bove the -is this mens tht when ou dd the -vlues, ou will obtin lrger positive -vlue. Look or regions where both grphs re negtive ie both lie below the -is this mens tht when ou dd the -vlues, ou will obtin more negtive -vlue. Consider the regions where the grphs dier in sign nd then be discerning in where the sum o the two vlues lie.. Look or smptotic behviour. I ou re sked to ind g, it is esier to sketch g,tht is, relect g in the - is nd continue s bove. ED Q,,,,,, 0,
Composite unctions Think o unction mchine, eg nd ind. IN domin + OUT rnge Wht hppens i we use mchines, eg nd g IN OUT + domin rnge h h or h g g h g g h is sid to be the composition o g with. h g or h g The domin o h or h is the domin o. Consider nd h. When =, = New unction hs been deined, h, Emple: Find both nd g :, g. Domin nge g 0 g g dom g g nd g, stting the domin nd rnge o ech, i :, g g dom g rn g 0, rn g, g is deined since rn g dom nd g is deined since rn dom g
Emple : I g, nd, 0 stte which o g nd g is deined b stte the domin nd rule o the deined. Domin nge 0 0 g rn g dom g is not deined rn dom g g is deined b g dom g dom 0 0 0 Emple : or nd g Is i g deined, ii g deined? b Determine restriction or, *, so tht g * is deined. Domin nge, g 0 0 0 = rn g dom g is deined rn dom g g is not deined 0 b For g dom * *: \ to bedeined rn g * g * dom g,, or \,,, * g dom g * dom * \, EE Q - e,,,,, 7,, 9, 0,, CAS Clcultor: composite unction: deine & g then g
Inverse Functions The Inverse o unction For the unction The grph o its inverse is ound b relecting the originl in the line. The rule o its inverse is ound b swpping or nd then mking the subject o the eqution. Emple: For the unction :[, ] where. Sketch the grph o ; b. Sketch the grph o ; c. Using the line s the mirror relect the grph o in it; d. Find the domin nd rnge or nd its inverse e. Find the rule or ;. Full deine. ; dom dom, rn,7,7 rn,
To ind the rule or the inverse we swp the nd in the originl eqution. swp or ull deined : :,7, The domin o = rnge - nd rnge = domin - I the grphs intersect, then the points o intersection MUST lso be on the line So the points o intersection cn be ound in ws: o o o. It is usull quicker nd esier to use one o the lst two. All unctions hve inverses, but the inverses m not be unctions the m onl be reltions. e.g. compre nd Originl is unction Originl is unction Inverse is unction Inverse is not unction A unction, hs n inverse unction, written onl i is one-to-one unction. i.e. horizontl nd verticl line onl crosses the grph o once. It is possible to restrict the domin on unction, so it will hve n inverse unction, e.g. the domin cn be restricted in mn ws, e.g { 0},,0,,0,,,
Emple: estrict the domin o g, so tht we hve n inverse unction g. Find the two possible g, where the domin is s lrge s possible. 0 g g is not : dom g rn g, 0 0 0 Let s choose the HS o the curve, i.e 0 0 0 0 0 0 0 g domin:, domin:, rnge: 0, rnge:,0 Let swp & which one is or g? As the dom g rn g & rn g dom g then g. g :,, g I we used the LHS o the curve, then g :,, g
Emple: I h : S, h, ind: S b h c h d h 7 S is the domin, dom h 0, nd rn h, b c h 9 h Let h swp & h :,, h d h 7 undeined, 7 is not in the domin, { 7}, Emple Find the inverse o the unction with rule nd sketch both unctions on one set o es, clerl showing the ect coordintes o intersection o the two grphs. Solution domin Swp & rn [, Note: The grph o is obtined b relecting the grph o in the line =. The grph o = is obtined rom the grph o = b ppling the trnsormtion,,. In this prticulr emple, it is simpler to solve = to solve the point o intersection. Grphicl Clcultor cn be used to ind the inverse o unction o Deine the unction o Solve =, EF Q,,,,, 7, ceg, 9, 0,,,,,
Power Functions p Power unctions re o the orm: ; p Q i.e. p is rtionl Strictl incresing nd strictl decresing unctions A unction is sid to be strictl incresing when < b implies <b or ll nd b in its domin. I unction is strictl incresing, then it is one-to-one unction nd hs n inverse tht is lso strictl incresing. Emple : The unction :, is strictl incresing with zero grdient t the origin. The inverse unction :,, is lso strictl incresing, with verticl tngent o undeined grdient t the origin. Emple : The hbrid unction g with domin [0, nd rule: 0 g is strictl incresing, nd is not dierentible t =. Emple : Consider h :, h H is not strictl incresing, But is strictl incresing over the intervl 0,. Strictl Decresing A unction is sid to be strictl decresing when < b implies b or ll nd b in its domin. A unction is sid to be strictl decresing over n intervl when < b implies b or ll nd b in its intervl. Emple : The unction :, The unction is strictl decresing over. e
Emple : The unction g :, g cos g is not strictl decresing. But g is strictl decresing over the intervl 0,. Power unctions with positive integer inde p Functions o the orm: ; p,,... groups: the even powers nd the odd powers. Even powers,,,,... o All hve the U-shped grph o Domin: o nge: { 0} or [0, o Strictl incresing or 0 o Strictl decresing or 0 o As, Odd powers,,,,... o All slope rom bottom let to top right o Domin: o nge: o Strictl incresing or ll o is one-to-one o As, &,
Power unctions with negtive integer inde p Functions o the orm: ; p,,... groups: the even powers nd the odd powers. Odd Negtive Powers p Functions o the orm: ; p,,... Sketch the grph o or Domin: \{0} nge: \{0} Asmptotes: o Horizontl: 0 As, 0 As, 0 o Verticl: 0 As 0, As 0, Odd unction: Sketch the grph o or
Even Negtive Powers p Functions o the orm: ; p,,... Sketch the grph o or Domin: \{0} nge: \{0} Asmptotes: o Horizontl: 0 As, 0 As, 0 o Verticl: 0 As 0, As 0, Even unction: Sketch the grph o or
Functions with rtionl powers: p q O the orm: emember Miml domin is [0, when q is even: q q q Miml domin is when q is odd: q q Consider: q Domin 0, i q is even nd \{0} i q is odd. : Asmptotes: 0 nd 0. q q In Generl: p p Alws deined or 0, nd when q is odd or ll. Eg: e.g.:
Inverses o Power Functions Emple: Find the inverse o ech o the ollowing:, : b,,0] : c, : d,, :,, :, :, rn or or &,, :, :, :[0,, : [0, rn or or &,,0] :, : dom Swp d c dom Swp b EG Q,,,,
Pst VCAA Em Questions 00 B D 009 B B
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