Transcription of Lecture 1 - UH
1 Lecture 1 Section One-To-One Functions; InversesJiwen He1 One-To-One Definition of the One-To-One FunctionsWhat are One-To-One Functions? Geometric TestHorizontal Line Test If some horizontal line intersects the graph of the function more than once,then the function is not one-to-one. If no horizontal line intersects the graph of the function more than once,then the function is are One-To-One Functions? Algebraic TestDefinition functionfis said to beone-to-one(or injective) iff(x1) =f(x2) impliesx1= functionfis one-to-one if and only if x1, x2,x16=x2impliesf(x1)6=f(x2).1 Examples and Counter-ExamplesExamples3. f(x) = 3x 5 is 1-to-1. f(x) =x2is not 1-to-1. f(x) =x3is 1-to-1. f(x) =1xis 1-to-1.
2 F(x) =xn x,n>0, is not f(x1) =f(x2) 3x1 5 = 3x2 5 x1=x2. Ingeneral,f(x) =ax b,a6= 0, is 1-to-1. f(1) = (1)2= 1 = ( 1)2=f( 1). In general ,f(x) =xn,neven, is not1-to-1. f(x1) =f(x2) x31=x32 x1=x2. In general ,f(x) =xn,nodd, is 1-to-1. f(x1) =f(x2) 1x1=1x2 x1=x2. In general ,f(x) =x n,nodd, is 1-to-1. f(0) = 0n 0 = 0 = (1)n 1 =f(1). In general , 1-to-1 offandgdoesnot always imply 1-to-1 off+ Properties of One-To-One FunctionsPropertiesPropertiesIffandgare one-to-one, thenf gis g(x1) =f g(x2) f(g(x1)) =f(g(x2)) g(x1) =g(x2) x1= f(x) = 3x3 5 is one-to-one, sincef=g uwhereg(u) =3u 5 andu(x) =x3are one-to-one. f(x) = (3x 5)3is one-to-one, sincef=g uwhereg(u) =u3andu(x) = 3x 5 are one-to-one.
3 F(x) =13x3 5is one-to-one, sincef=g uwhereg(u) =1uandu(x) =3x3 5 are Increasing/Decreasing Functions and One-To-OnenessIncreasing/Decreasing Functions and One-To-OnenessDefinition 5. A functionfis (strictly)increasingif x1, x2,x1<x2impliesf(x1)<f(x2). A functionfis (strictly)decreasingif x1, x2,x1<x2impliesf(x1)>f(x2).Theorem that are increasing or decreasing are , eitherx1<x2orx1>x2ans so, by monotonicity, eitherf(x1)<f(x2) orf(x1)>f(x2), thusf(x1)6=f(x2).Sign of the Derivative Test for One-To-OnenessTheorem 7. Iff (x)>0for allx, thenfis increasing, thus one-to-one. Iff (x)<0for allx, thenfis decreasing, thus f(x) =x3+12xis one-to-one, sincef (x) = 3x2+12>0 for allx. f(x) = x5 2x3 2xis one-to-one, sincef (x) = 5x4 6x2 2<0 for allx.
4 F(x) =x +cosxis one-to-one, sincef (x) = 1 sinx 0andf (x) = 0 only atx= 2+ 2k .2 Inverse Definition of Inverse FunctionsWhat are Inverse Functions?3 Definition a one-to-one function. Theinverseoff, denoted byf 1, is the unique function with domain equal to the range offthat satisfiesf(f 1(x))=xfor allxin the range T Confusef 1with the reciprocal off, that is, with 1/f. The 1 in the notation for the inverse offisnot an exponent;f 1(x)does not mean1/f(x).Example4 Example10. f(x) =x3 f 1(x) =x1 By definition,f 1satisfies the equationf(f 1(x))=xfor allx. Sety=f 1(x) and solvef(y) =xfory:f(y) =x y3=x y=x1/3. Substitutef 1(x) back in fory,f 1(x) =x1 general ,f(x) =xn,nodd, f 1(x) =x1 f(x) = 3x 5 f 1(x) =13x+ By definition,f 1satisfiesf(f 1(x))=x, x.
5 Sety=f 1(x) and solvef(y) =xfory:f(y) =x 3y 5 =x y=13x+53. Substitutef 1(x) back in fory,f 1(x) =13x+ general ,f(x) =ax+b,a6= 0, f 1(x) =1ax Properties of Inverse FunctionsUndone Propertiesf f 1= IdR(f)D(f 1) =R(f)8f 1 f= IdD(f)R(f 1) =D(f)Theorem definition,f 1satisfiesf(f 1(x))=xfor allxin the range is also true thatf 1(f(x))=xfor allxin the domain x D(f), sety=f(x). Sincey R(f),f(f 1(y))=y f(f 1(f(x)))=f(x). fbeing one-to-one impliesf 1(f(x)))= offandf 1910 Graphs offandf 1 The graph off 1is the graph offreflected in the liney= the graph off, sketch the graph off draw the liney=x. Then reflect the graph offin that continuous so isf Differentiability of InversesDifferentiability of InversesTheorem 15.
6 (f 1) (y) =1f (x),f (x)6= 0,y=f(x).Proof. y D(f 1) =R(f), x D(f) (x). By definition,f 1(f(x)) =x ddxf 1(f(x)) = (f 1) (f(x))f (x) = Iff (x)6= 0, then(f 1) (f(x)) =1f (x) (f 1) (y) =1f (x). (x) =x3+12x. Calculate (f 1) (9).Solution Note thatf (x) = 3x2+12>0, thusfis one-to-one. Note that (f 1) (y) =1f (x),y=f(x). To calculate (f 1) (y) aty= 9, find a (x) = 9:f(x) = 9 x3+12x= 9 x= 2. Sincef (2) = 3(2)2+12=252, then (f 1) (9) =1f (2)= that to calculate (f 1) (y) at a specificyusing(f 1) (y) =1f (x),f (x)6= 0,y=f(x),we only need the value (x) =y, not the inverse functionf 1, whichmay not be known GradesDaily (x) =x,f 1(x) =? :(a) not exist, (b)x,(c) (x) =x3,f 1(x) =? :(a) not exist, (b)x13, (c) (x) =x2,f 1(x) =?
7 :(a) not exist, (b)x12, (c) (x) = 3x 3,(f 1) (1) =? : (a) not exist, (b) 3,(c) One-To-One Definition .. Properties .. Monotonicity ..32 Definition .. Properties .. Differentiability ..1113
