1. Inverse Functions Undoing What Functions Do

2. 6/1/2013 Inverse Functions 2 One-to-One Functions Definition A function f is a one-to-one function if no two ordered pairs of f have the same second component Note: One-to-one is often written as 1-1

3. 6/1/2013 Inverse Functions 3 One-to-One Functions 1-1 Examples: 1. f = { (1, 3), (2, 5), (3, 2), (7, 1) } 2. g = { (1, 3), (2, 5), (3, 6), (7, 3) } 3. h = { (5, 3), (2, 9), (5, 6), (8, 7) } 4. f(x) = (x – 4) 2 + 7 5. g(x) = x + 1 6. f(x) = |x + 1| Inverse Functions NOT 1-1 NOT a function WHY ?

4. 6/1/2013 Inverse Functions 4 Horizontal Line Test No horizontal line intersects the graph of a 1-1 function more than once Examples x y(x) x ● ● ● 1-1 function Not 1-1

5. 6/1/2013 Inverse Functions 5 Horizontal Line Test No horizontal line intersects the graph of a 1-1 function more than once More Examples x y(x) x x ●● ● ● ● ● ● ● NOT a function ! Not 1-1

6. 6/1/2013 Inverse Functions 6 The Inverse of a Function Example: Let y = f(x) = 2x + 1 Solving for x : x = (1/2)(y – 1) = g(y) Each x is mapped to a particular y by f(x), and that y is mapped back to the original x by g(y) So, if x = 3, then f(3) = 2(3) + 1 = 7 and g(7) = (1/2)(7 – 1) = 3

7. 6/1/2013 Inverse Functions 7 The Inverse of a Function If x = 3, then f(3) = 2(3) + 1 = 7 3 7 f g and g(7) = (1/2)(7 – 1) = 3 Domain f Domain g Questions: What are g is the inverse of f ? (g f)(x) = g(f(x)) (f g)(y) = f(g(y)) ?

8. 6/1/2013 Inverse Functions 8 Inverse Notation If function g is the inverse function for function f we write this as g = f –1 Note: f –1 does NOT mean the reciprocal of f That is f –1 (x) ≠ Inverse Functions f(x) 1

9. 6/1/2013 Inverse Functions 9 Inverse Notation If function g is the inverse function for function f we write this as g = f –1 Questions Inverse Functions What is (f f –1 )(x) ? (f –1 f)(x) ? What is Does every function have an inverse ? If not, what guarantees an inverse ?

10. 6/1/2013 Inverse Functions 10 Definition A 1-1 function f has inverse f –1 Inverse Functions (f –1 f)(x) = f –1 (f(x)) = x for every x in the domain of f for every x in the domain of f –1 (f f –1 )(x) = f(f –1 (x)) = x AND IF and ONLY IF

11. 6/1/2013 Inverse Functions 11 Definition A 1-1 function f has inverse f –1 Inverse Functions (f –1 f)(x) = f –1 (f(x)) = x for every x in the domain of f for every x in the domain of f –1 Note: The name of variable x is a dummy name … both can use x Since f and f –1 are different functions IF and ONLY IF AND (f f –1 )(x) = f(f –1 (x)) = x

12. 6/1/2013 Inverse Functions 12 Inverse Functions Example: For f(x) = 2x + 1 and f –1 (x) = (1/2)(x – 1) we have (f –1 f)(x) = f –1 (2x + 1) = (1/2)((2x + 1) – 1) = x x y f(x) = 2x + 1 f –1 (x) = (x – 1)/2 L1L1 L2L2 = f –1 (f(x))

13. 6/1/2013 Inverse Functions 13 Inverse Functions Example: f(x) = 2x + 1 and f –1 (x) = (1/2)(x – 1) k 2k + 1 k y = x (2k + 1, k) (k, 2k + 1) x y f(x) = 2x + 1 f –1 (x) = (x – 1)/2 L1L1 L2L2 ● ● ● ● ● ● Each point on L 2... and conversely corresponding point on L 1 Pick an x = k … and follow it Feed 2k + 1 to f –1 … to return k is a reflection across line y = x of a

14. 6/1/2013 Inverse Functions 14 Inverse Functions and Graphs For any 1-1 function f(x) with graph { (x, y)  y = f(x) } the inverse function f –1 has a graph { (y, x)  x = f –1 (y) } which is a reflection of the graph of f(x) through the line y = x x y f(k) k k y = f(x) y = f –1 (x) y = x (k, f(k)) (f(k), k) ● ● ● ● ● ●

15. 6/1/2013 Inverse Functions 15 x y f(k) k k y = f(x) y = f –1 (x) y = x (k, f(k)) (f(k), k) ● ● ● ● ● ● Inverse Functions and Graphs 1-1 function f(x) with inverse function f –1 Note that y = f(x) so we have x = f –1 (y) Interchanging the names of x and y yields the graph of f –1 as { (x, y)  y = f –1 (x) }

16. 6/1/2013 Inverse Functions 16 Inverse Functions Inverse Function Fact A function f has an inverse function f –1 IF and ONLY IF Example 1. f = { (3, 15), (4, 10), (5, 7), (6, 4), (7, 3) } f –1 = { (15, 3), (10, 4), (7, 5), (4, 6), (3, 7) } (f –1 f)(4) f –1 (f(4)) = f –1 (10) = 4 = f is a 1-1 function Thus

17. 6/1/2013 Inverse Functions 17 Inverse Functions Example 1. f = { (3, 15), (4, 10), (5, 7), (6, 4), (7, 3) } f –1 = { (15, 3), (10, 4), (7, 5), (4, 6), (3, 7) } In General : (f f –1 )(4) ? What about Does 4 have to be in the domain of f –1 ? = (f –1 f)(4) f –1 (f(4)) = f –1 (10) 4 = Thus Question: (f –1 f)(x) f –1 (f(x)) = x =

18. 6/1/2013 Inverse Functions 18 Inverse Functions Example 2. (f f –1 )(2) ? What about f = { (x, y) | y = } x – 4  + 7 … for x ≥ 4 Is this 1-1 ? Well … … is it ? NOTE: … if y ≥ 7 ? … so that y ≥ 7 … 2 would have to be in the range of f f –1 = { (y, x) | x = (y – 7) 2 + 4 } (f –1 f)(4) f –1 (f(4)) = 4 = … if y < 7 ? f(x) ≥ 7 > 2 for all x ≥ 4

19. 6/1/2013 Inverse Functions 19 Inverse Functions Example 2. f = { (x, y) | y = } x – 4  + 7 f –1 = { (y, x) | x = (y – 7) 2 + 4 } Question: What are the domain and range of f ? What about f –1 ? Dom f = { x  x ≥ 4 } = [ 4,  ) Range f = { x  x ≥ 7 } = [ 7,  ) Dom f –1 = { x  x ≥ 7 } = [ 7,  ) Range f –1 = { x  x ≥ 4 } = [ 4,  )

20. 6/1/2013 Inverse Functions 20 Examples: Find the inverses where they exist 3. g(x) = Inverse Functions x + 5  + 1 Domain ? g –1 (x) = Range ? [ 1,  ) [ –5,  ) (x – 1) 2 – 5 Domain ? Range ? [ –5,  ) [ 1,  )

21. 6/1/2013 Inverse Functions 21 Examples: Find the inverses where they exist 4. h(t) 5. y = | x + 1 | Inverse Functions Domain ? Range ? h –1 (t) = ( – ,  ) [ 6,  ) NONE 1 2 t 2 + 6 = h is not 1-1 so has no inverse Domain ? Range ? y –1 (x) = NONE ( – ,  ) [ –1,  ) y is not 1-1 so has no inverse

22. 6/1/2013 Inverse Functions 22 Inverse Functions Example: 6. Find the graph of the inverse function if it exists Identify intercepts Plot y = x line Find intercept reflections x y (0,–5)  (2, 0)  y = f(x)   (–5, 0) (0, 2) y = x

23. 6/1/2013 Inverse Functions 23 Inverse Functions Example: 6. Find the graph of the inverse function Plot the inverse graph x y (0,–5)  (2, 0)  y = f(x)   (–5, 0) (0, 2) y = f –1 (x) y = x Question: What are the equations of the graphs of f and f –1 ?

24. 6/1/2013 Inverse Functions 24 Inverse Functions x y (0,–5)  (2, 0)  y = f(x)   (–5, 0) (0, 2) y = f –1 (x) y = x Question: What are the equations of the graphs of f and f –1 ? 5 2 y = f(x) = x – 5 Use intercepts for slopes: y = f –1 (x) = x + 2 2 5

25. 6/1/2013 Inverse Functions 25 Inverse Functions Example: 7. Find the graph of the inverse function if it exists Identify intercepts Plot y = x line Find intercept reflections x y  y = f(x)  (3, 0) (0, 3) y = x (0, 6) (6, 0)  

26. 6/1/2013 Inverse Functions 26 Inverse Functions Example: 7. Find the graph of the inverse function Plot the inverse graph x y  y = f(x)  (3, 0) (0, 3) y = f –1 (x) y = x (0, 6) (6, 0)   Question: What are the equations of the graphs of f and f –1 ?

27. 6/1/2013 Inverse Functions 27 Inverse Functions x y  y = f(x)  (3, 0) (0, 3) y = f –1 (x) y = x (0, 6) (6, 0)   Question: What are the equations of the graphs of f and f –1 ? Use intercepts for slopes: y = f –1 (x) = –2x + 6 y = f(x) = x + 3 1 2 –

28. 6/1/2013 Inverse Functions 28 Think about it !