F(x) X #-3x Ñ Precomposing Equations F O G(x) F O Call G .

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2 xy 2. Let’s precompose p with the additionSupposep(x, 3xSupposep(x,y)y)p(x, 23x 3xxy2.xy Let’sprecomposep withtheSuppose2Supposep(x,2 dditionadditionp pwithy)y) 2.2. y).pfunctionA( 3,4)A(.A(Thatis,Thatwe’llpfind findA( 3,4)y). 3(x,)4,3.Thatwe’ll)4,3A((x,p po eformulaforpp ulaformulaforp p32xx 22x3-x 22x3x esof tesof t)4,3A(replacesofofof(x,(x,y)y)( eswith.vectorwith.(x,y) with.with.r)r)r) (i-3d (i-3d(i-3dx’ ingwitheachxeachy)Third,Third,the formulaformulafor p oforAA(y)isisfoundbyrewriting acing( 3,4)formulafor)4,3(x,foieplacingwithx xwitheachy)y)isisp po (x,xp, exceptreplacing3, xandeachwith4.yy thatwe’llreplaceeachxwithx 3,andreplaceeachywithy 4.x 3,3,andandreplacingreplacingeacheachy ywithwithy y 4.4.———(x3)23 (x3)2 2 2(x-3)(r)33(x3)2 simplified.&-I4tt}3)223)2)4,3(x,3(x4)3)(y3)(yy)p0 AA((x,A(y) 3(x (x(x 3)23)(y 4)3)(y 2 2 4)4) 2 200( 3,4))A(4,3(x,)4,3(x,(x(x 3(x3(xy) y) 3)—————————223x18x 2—xy—4x 3y 2i-2 3x 18x 2718x 2—xy—4x 3y xy 4x 3y 12 2 2i-22 3x23x18x 2—xy—4x 3y 2i-222 xyxy22x 3x3x 3y3y 412xyxy22x 3x222x3x22x 3y3y ———————********——******** * ** * ** * ** * ** * ** * ** * ** * ** * ** * ** * ** *126126126126**

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As we saw in the previous chapter, if T : R2 R2 is a planar transformaAs andwe sawthe setpreviouschapter,if Ty): q(x,—* Ra planartion,if S inis theof solutionsof p(x,y),isthenT (S) transformais the set of 1 1 y) q(x, y), then T(S) is the set oftion,andifisthesetofsolutionsSofp(x,solutions of p(x, y) T q(x, y) T . We can say this more economicallysolutionsasfollows:of p(x, y) oT q(x, y) oT. We can say this more economicallyas follows:The equation for S precomposed with T 1The equation for S precomposedwith TforeachT (S).Third, rewrite the equation isforan5,equationreplacingx with x y and each yis an equation for T(S).with —x 2y.—To precompose an equation with T 1 , there are three steps to be followed.To precompose an equation with T’, there are three steps to be followed.Step 1: Write the original equation.Step 1: Write the original equation.Step 2: Write what T 1 replaces each of the coordinates of the vectorStep 2: (x,Writehow T’ replaces the coordinates of the vector (x, y).y) with.3: RewriteStep 3:Rewrite thethe equationequation fromfrom Step1, exceptexcept replaceStep 1,replace everyevery xx andandStepevery yy withwith thethe formulasformulas identifiedidentified inin StepStep 2.2.everyWe’ll practicepractice thesethese threethree steps–practicesteps—practice precomposingprecomposing equations–withequations—with theWe’llthe(-i 2-(x-) iQnextproblem.next problem.(x-(-x2) 3Problem: SupposeSuppose thatthat SS isis thethe subsetsubset ofof thethe planeplane thatthat isis thethe setset ofofProblem: solutions ofof thethe equationequation xyxy 22 xx 4y10. 10. 33 4ysolutions—Now simplify the equation so that the answer, an equation that has N(S)Sas its set of solutions, is 10 Let N be the invertible matrix N Let N be the invertible matrix N 2 1. Give an equation that1 1Give an equation that( ).128128

N(S)is theof.set of so1utions.N(S)theofsolutionsN (S) isisN(S)the setsetofsolutionsof.is the set of so1utions.N(5)find anforequationforN(S),have to ution:Solution:To findfind ananToequationforN (S), wewe havehave totoweprecomposethe equaequa- the equaan equationfor N(S), we have to precompose the equafor 1S .with.1NtionforwithN’tionSolution:for S tionwithToNfindtion for S with N.1down theforequationfor S.First,downtheS.First, writewriteFirst,downwritethe equationequationforS.First, write down the equation for S. 3 3 10 10 1 down how NSecond,1ofreplacesthe coordinatesof (x,(x,Second, ritehowN’ I-I-I /i -I2(I-I /i(Iz(i)—i(i)—I 2) -Iz(i)—i(i)—I 2)2N--N1‘V(\ 2)‘VN1(/j)2\ 2)(i(i—1—1/j)2129129129129

Third, rewrite the equation for 5, replacing each x with xwith —x 2y.—y and each yThird, rewrite the equation for S, except replace each x with x y andeach y with x 2y.(x-(-x2) 3(-i 2-(x-) iQNow simplify the equation so that the answer, an equation that has N(S)as its set of solutions, isNow simplify the equation so that the answer–an equation that has N (S)as its set of solutions–is x2 3xy 2y 2 3 4x2 16xy 16y 2 x y 10131130

ExercisesExercisesForFor #1-8,#1-8, writewrite allall polynomialspolynomials inin thethe formform Ax 2Bxy CyAx2 Dx Ey F Bxy Cy 2 Dx Ey F1.) Precompose).121.)Precompose thethe equationequation xx yy 33 yy 44 withwith A(A(1,2) .——2.)2)3,2.22.) PrecomposePrecompose thethe equationequation 2x2x2 3xy3xy yy 11 2xy2xy yy 2 withwith A(A( 2,3) 303.)3.) PrecomposePrecompose thethe equatioiiequation xx 22 xyxy yy 11 withwith.2 1 13). 114.)224.) PrecomposePrecompose thethe equationequation xx2 xx 3y 2y2y 2 5y5y 22 withwith (21.2 3——( ).———ExercisesForFor #5-8,#5-8, supposesuppose thatthat SS isis thethe setset ofof solutionssolutions ofof thethe equationequation923xFor #1-8, write all polynomialsin xythe form3x2 xy y x—y 5 y2 x y 52 Bxy CyAx2 Dx Ey F1.) Precompose the equation x—2.) Precompose the equation 2x2y 3—3.) Precompose the equation x 2 y3xy y xy———y4 with1— ).232 with A(xy y21 with(30211equation2 x for S precoinposedx(214.) Precomposethethat22y2withwithT’5y 1For #5-8,usetheequationisan#5-8, use that thefor precomposed with T is anequation—3equationfor T(S).T (S).For #5-8, suppose that S is the set of solutions of the equation)2? Whatis5.) What5.)What isis AA 1Whatis the9 equation forfor A(3,2) (S)?2(3,2) ?31 xy y x—y 5—7—M(s)Z(5)3A(--131131For #5-8, use that the equation for S precoinposed with T’ is an equationfor T(S).)? What is the equation for A(95.) ‘Vhat is A’)(S)?32

For #5-8, use that the equation for S precoinposed with T’ is an equationfor T(S).4.) Precompose the equation x2 x 2y2 5y 2 with (2113).—)? What is the equation for A(95.) ‘Vhat is A’)(S)?32For #5-8, suppose that S is the set of solutions of the equationis the equationfor A(S)?6.) What is A 19(2, 4) ? What3x 2 xy y x—y 5(2, 4)6For #5-8, use that the equation for S precoinposed with T’ is an equation for T(S).1 4. What is M 1 ? What is the equation for M (S)?7.) Suppose M 01For Whatall polynomialsthe form#1-8, iswrite)?2AWhat is the inequationfor5.) Dx Ey FAx 2Bxy CyExercises1.) Precompose the equation x—2.) Precompose the equation 2x27).2y 4 with A(i,M(s)y 3— —3xy y—1 2xy y).3,22 with A(--3.) Precompose the equation x 2 4.) Precompose the equation 2x x131—— y—22y1 with—( ).5y 2 with (211—3 For #5-8, suppose that3 5 S is the set of 1solutions of the equation8.) Suppose N . What is N ? What is the equation for N (S)?1 232 y y2 x y 5—BN(S)For #5-8, use that the equation for S precomposed with T’ is an equationfor T(S).1325.) What isWhat is the equation for

x 1f (x) x2 57if x ( , 0);if x [0, 4]; andif x (4, ).Find the following values.9.) f ( 2)12.) f (1)15.) f (4)10.) f ( 1)13.) f (2)16.) f (5)11.) f (0)14.) f (3)17.) f (6)Multiply the following matrices. 2 43 418.)1 10 2 0 12 219.)2 31 3Solve the following equations.20.) loge (x)2 5 loge (x) 6 021.) (x3 3x2 2x 3)2 1133

every ywith the formulas identi ed in Step 2. We’ll practice these three steps{practice precomposing equations{with the next problem. Problem: Suppose that S is the subset of the plane that is the set of solutions of the eq