1. (CSC 102) Lecture 21 Discrete Structures

2. Previous Lecture Summery  Sum/Difference of Two Functions  Equality of Two Functions  One-to-One Function  Onto Function  Bijective Function (One-to-One correspondence)

3. Today’s Lecture  Inverse Functions  Finding an Inverse Function  Composition of Functions  Composition of Functions defined on finite sets  Plotting Functions

4. Inverse Functions Theorem The function F -1 is called inverse function.

5. Inverse Functions Given an arrow diagram for a function. Draw the arrow diagram for the inverse of this function

6. Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x

7. Theorem

8. Composition of Functions

9. Composition of Functions defined on finite sets Let X = {1, 2, 3}, Y ’ = {a, b, c, d}, Y = {a, b, c, d, e}, and Z = {x, y, z}. Define functions f : X → Y’ and g: Y → Z by the arrow diagrams below. Draw the arrow diagram for g ◦ f. What is the range of g ◦ f ?

10. Composition of Functions defined on finite sets

11. To find the arrow diagram for g ◦ f, just trace the arrows all the way across from X to Z through Y. The result is shown below. Composition of Functions defined on finite sets Let X = {1, 2, 3}, Y ’ = {a, b, c, d}, Y = {a, b, c, d, e}, and Z = {x, y, z}. The range of g ◦ f is {y, z}.

12. Composition of Functions defined on Infinite Sets Let f : Z → Z, and g: Z → Z be two functions. s.t., f (n)=n + 1 for all n ∈ Z and g(n) = n 2 for all n ∈ Z. a. Find the compositions g◦f and f◦g. b. Is g ◦ f = f ◦g? Explain. a.The functions g ◦ f and f ◦g are defined as follows: (g ◦ f )(n) = g( f (n)) = g(n + 1) = (n + 1) 2 for all n ∈ Z, ( f ◦g)(n) = f (g(n)) = f (n 2 ) = n 2 + 1 for all n ∈ Z.

13. Composition of Functions defined on Infinite Sets Let f : Z → Z, and g: Z → Z be two functions. s.t., f (n)=n + 1 for all n ∈ Z and g(n) = n 2 for all n ∈ Z. b. Is g ◦ f = f ◦g? Explain.

14. Composition with Identity Function Let X = {a, b, c, d} and Y = {u, v,w}, and suppose f : X → Y is given by the arrow diagram Find f ◦ I X and I Y ◦ f.

15. Composition with Identity Function

16. Composing a Function with its Inverse

17. In a similar way, we can show that Composing a Function with its Inverse

18. Composition of One-to-One Functions

25. Domain =Range =

27. Lecture Summery  One-to-One Function  Onto Function  Bijective Function (One-to-One correspondence)  Inverse Functions