One-to-One and Inverse Functions. Learning Objectives. Properties of Functions

Transcription

1 One-to-One and Inverse Functions Learning Objectives. Determine whether a unction is one-to-one 2. Determine the inverse o a unction deined by a map or a set o ordered pairs 3. Obtain the graph o the inverse unction rom the graph o the unction 4. Find the inverse o a unction deined by an equation 2 A unction :AB is said to be one-to-one (or injective), i and only i and y A (() = (y) = y) In other words: is one-to-one i and only i it does not map two distinct elements o A onto the same element o B 3

2 Eample (Maria) = San Antonio (Juan) = El Paso (Lisa) = Austin (Peter) = El Paso Is one-to-one? No, Juan and Peter are mapped onto the same element o the image 4 Eample g(jannie) = San Antonio g(juan) = El Paso g(lisa) = Austin g(peter) = Lubbock Is g one-to-one? Yes, each element is assigned a unique element o the image. 5 Theorem Horizontal Line Test I horizontal lines intersect the graph o a unction in at most one point, then is one-to-one. 6 2

3 Eample Use the graph to determine whether the ollowing unction is one - to - one. y ( ) Yes, () passes the horizontal line test 7 Eample ( ) 2 2 Use the graph to determine whether the ollowing unction is one - to - one. y No, () ails the horizontal line test 8 Eample Which o the ollowing are one - to - one unctions? {(, ), (2, 4), (3, 9), (4, 6)} one-to-one {(-2, 4), (-, ), (0, 0), (, )} not one-to-one 9 3

4 Eample Prove :RR with () = 3 is one-to-one One-to-one means, ya (() = (y) = y) To show: () (y) whenever y (indirect proo) y 3 3y () (y), so i y, then () (y), that is, is one-to-one 0 A unction :AB with A,B R is strictly increasing, i,ya ( < y () < (y)) strictly decreasing, i,ya ( < y () > (y)) A unction that is either strictly increasing or strictly decreasing is one-to-one We can use this to prove a unction is to one-toone by showing the slope is always positive, or always negative A unction :AB is called onto, or surjective, i and only i or every element bb there is an element aa with (a) = b In other words, is onto i and only i its range is its entire codomain In other words a unction is onto i we use all o the y-values 2 4

5 A unction : AB is a bijection, i and only i it is both one-to-one and onto I is a bijection and A and B are inite sets, then A = B 3 Maria Hector Lisa Jon El Paso Hondo Austin San Antonio Is injective? Is surjective? Is bijective? 4 Maria Paul Lisa Liz Paul El Paso Hondo Austin San Antonio Is injective? Is surjective? Yes. Is bijective? 5 5

6 Maria Jarvis Lisa Doug El Paso Hondo Austin Tyler Amarillo Is injective? Yes. Is surjective? Is bijective? 6 Maria Alberto Lisa Clyde El Paso Dallas Austin San Antonio Houston Is injective? No! is not even a unction! 7 Maria Edward Lisa Peter Olga El Paso La Vernia Austin San Antonio Laredo Is injective? Yes. Is surjective? Yes. Is bijective? Yes. 8 6

7 Inverse Function Injective (one-to-one) unctions have an inverse unction The inverse unction o the one-to-one :AB is the unction - :BA with - (b) = a whenever (a) = b 9 Inverse Function Domain o Range o Range o - Domain o - Domain o Range o Range o Domain o 20 Eample (Maria) = San Antonio (Robert) = El Paso (Patti) = Austin (Peter) = Pearsall (Olga) = Hondo Clearly, is bijective hence one-to-one. The inverse unction - is given by: - (San Antonio) = Maria - (El Paso) = Robert - (Austin) = Patti - (Pearsall) = Peter - (Hondo) = Olga 2 7

8 Composition The composition o two unctions g:ab and :BC, denoted by g g a g a This means that irst, unction g is applied to element aa, mapping it onto an element o B, then, unction is applied to this element o B, mapping it onto an element o C. Thereore, the composite unction maps rom A to C. 22 Composition g A g B C a b c g (a) = b (b) = c ( g(a) ) = c 23 Property o Inverse Function a a b b a We can use this to veriy unctions are inverse b 24 8

9 Eample Find the inverse 3, 3, 2, 7, 2,4, 3,4 3, 3, 7, 2, 4,2, 4,3 25 Drawing Inverse Function 26 Remarks The inverse o a -to- unction is a symmetric with the line y = This can be used to veriy we have correctly ound the inverse o a unction 27 9

10 Finding the Inverse Function ) Veriy is -to- 2) Replace () with y 3) Swap and y 4) Solve or y 5) Replace y with - () 28 Find the inverse 3 This is (translated to right 3 units) is -to- y y y y 3 Eample y3 y Eample Show unctions are inverses 3 g 3 g g

11 Eample Replace g() with y Swap and y Find the inverse y y y g Solve or y y Replace y with g - () y y y y y g y y 3