5.1: Inverse Functions Dr. Kim University of Wisconsin-La Crosse - PowerPoint PPT Presentation

5.1: Inverse Functions Dr. Kim University of Wisconsin-La Crosse Fall 2014 1

Question 1 Let’s talk about what happens to the axes after doing “the flip”. After performing “the flip”, A The x -axis (which was horizontal) becomes vertical and vice versa, and I’m very sure. B The x -axis (which was horizontal) becomes vertical and vice versa, but I’m not sure. C The x -axis (which was horizontal) becomes stays horizontal, and I’m very sure. D The x -axis (which was horizontal) becomes stays horizontal, but I’m not sure. 2

Question 1 Let’s talk about what happens to the axes after doing “the flip”. After performing “the flip”, A The x -axis (which was horizontal) becomes vertical and vice versa, and I’m very sure. � B The x -axis (which was horizontal) becomes vertical and vice versa, but I’m not sure. � C The x -axis (which was horizontal) becomes stays horizontal, and I’m very sure. D The x -axis (which was horizontal) becomes stays horizontal, but I’m not sure. 2

Question 2 Draw the point at ( x , y ) = ( 10 , − 1 ) . This point is “mostly east” and “slightly south”. (Feel free to draw the point on your transparency, but only do the “flip” MENTALLY.) After performing “the flip”, the new point will be at A ( 10 , − 1 ) , so stays in same location B ( − 1 , 10 ) , thus mostly north and slightly west C ( − 10 , 1 ) , thus mostly west and slightly north D ( 1 , − 10 ) , thus mostly south and slightly east E The point ends up somewhere else 3

Question 2 Draw the point at ( x , y ) = ( 10 , − 1 ) . This point is “mostly east” and “slightly south”. (Feel free to draw the point on your transparency, but only do the “flip” MENTALLY.) After performing “the flip”, the new point will be at A ( 10 , − 1 ) , so stays in same location B ( − 1 , 10 ) , thus mostly north and slightly west � C ( − 10 , 1 ) , thus mostly west and slightly north D ( 1 , − 10 ) , thus mostly south and slightly east E The point ends up somewhere else 3

Question 3 After performing “the flip”, a point with a big x -value becomes A a point with a small x -value B a point with a big x -value C a point with a small y -value D a point with a big y -value E something else happens 4

Question 3 After performing “the flip”, a point with a big x -value becomes A a point with a small x -value B a point with a big x -value C a point with a small y -value D a point with a big y -value � E something else happens 4

Question 4 Go back to “pre-flip” setting. Draw a horizontal line. After applying the flip, A the line will appear horizontal, and I’m very sure. B the line will appear horizontal, but I’m not sure. C the line becomes vertical, and I’m very sure. D the line becomes vertical, but I’m not sure. 5

Question 4 Go back to “pre-flip” setting. Draw a horizontal line. After applying the flip, A the line will appear horizontal, and I’m very sure. B the line will appear horizontal, but I’m not sure. C the line becomes vertical, and I’m very sure. � D the line becomes vertical, but I’m not sure. � 5

Arrow diagrams 1 A 2 B 3 C 4 D 6

Arrow diagrams 1 A 2 B 3 C 4 D inputs outputs Function because each input knows who to throw the ball to! 6

Arrow diagrams 1 A 2 B 3 C 4 D inputs outputs but NOT one-to-one because output B receives the ball MORE than once! (HLT fails) 6

Arrow diagrams 1 A 2 B 3 C 4 D inputs outputs What if we switch all the arrows? 6

Arrow diagrams 1 A 2 B 3 C 4 D outputs inputs 6

Arrow diagrams 1 A 2 B 3 C 4 D outputs inputs Now input B has too many people to throw the ball to. With the arrows pointing the other way, this is NOT a function! (VLT fails) 6

Why inverse functions? 7

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 5 . 29 , so this is the cost with 1 topping. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 5 . 29 , so this is the cost with 1 topping. Example: f ( 2 ) = 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 5 . 29 , so this is the cost with 1 topping. Example: f ( 2 ) = 5 . 79 , so this is the cost with 2 toppings. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 5 . 29 , so this is the cost with 1 topping. Example: f ( 2 ) = 5 . 79 , so this is the cost with 2 toppings. Input is number of toppings. Output is price. 8

Why inverse functions? Culver’s Concrete Mixers!!! The base concrete mixer starts at $4.79. Each topping costs $0.50. Use x as input, which will represent the NUMBER of toppings ordered. f ( x ) = 4 . 79 + 0 . 50 x is the cost. Example: f ( 0 ) = 4 . 79 , so this is the cost with no toppings. Example: f ( 1 ) = 5 . 29 , so this is the cost with 1 topping. Example: f ( 2 ) = 5 . 79 , so this is the cost with 2 toppings. Input is number of toppings. Output is price. What’s the inverse function? Helps answer questions like: If I have $14.79, then how many toppings can I have? Input is price. Output is number of toppings. 8