1. Section 2.3

2. Section Summary  Definition of a Function. o Domain, Cdomain o Image, Preimage  One-to-one (Injection), onto (Surjection), Bijection  Inverse Function  Function Composition  Graphing Functions  Floor, Ceiling, Factorial

3. Functions  Definition 1: Let A and B be nonempty sets. A function f from A to B, denoted f: A → B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.  Functions are sometimes called mappings or transformations. A B C StudentsGrades D F Kathy Scott Sandeep Patel Carlota Rodriguez Jalen Williams

4. Functions  Given a function f: A → B: o We say f maps A to B or f is a mapping from A to B. o A is called the domain of f. o B is called the codomain of f.  If f(a) = b, then b is called the image of a under f. a is called the preimage of b.  The range of f is the set of all images of points in A under f. We denote it by f(A).  Two functions are equal when they have the same domain, the same codomain and map each element of the domain to the same element of the codomain.

5. Example  f(a) = ? ozoz  The image of d is ? ozoz  The domain of f is ? oAoA  The codomain of f is ? o B  The preimage of y is ? obob  f(A) = ? o {y,z}  The preimage(s) of z is (are) ? o {a,c,d} AB a b c d x y z

6. One-to-one function  Definition 5: A function f is said to be one-to-one, or injunction, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f.  Example: Determine whether the function f from {a, b, c, d} to {1, 2, 3, 4, 5} with f (a) = 4, f (b) = 5, f (c) = 1, and f (d) = 3 is one- to-one.  Solution : yes.

7. Onto function  Definition 7: A function f from A to B is called onto or surjection, if and only if for every element there is an element with.  Example: Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f (a) = 3, f (b) = 2, f (c) = 1, and f (d) = 3. Is f an onto function?  Solution: Yes.

8. Bijections  Definition 8: A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjection and injection).

9. Composition  Definition 10: Let g : A → B and let f :B → C. The composition of the functions f and g, denoted for all a ∈ A by f o g, is defined by (f o g)(a) = f (g(a)).

10. Composition AC a b c d i j h AB C a b c d V W X Y g h j i f

11. Example 1 : If and, then and

12. Composition Questions  Example 2: Let f and g be functions from the set of integers to the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2.  What is the composition of f and g, and also the composition of g and f ?  Solution: o f ∘g ( x )= f (g( x )) = f (3 x + 2) = 2(3 x + 2) + 3 = 6 x + 7 o g ∘f ( x )= g (f( x )) = g (2 x + 3) = 3(2 x + 3) + 2 = 6 x + 11

13. Graphs of Functions  Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs {( a,b ) | a ∈A and f ( a ) = b }.

14. Some Important Functions  Definition 12:  The floor function assigns to the real number x the largest integer that is less than or equal to x. The value of the floor function at x is denoted by.  The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x. The value of the ceiling function at x is denoted by.  Example:

15. Floor and Ceiling Functions