E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 2.3 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Functions KFUPM - Prep Year Math Program (c) All Right Reserved  Definition of a Function  Domain of a Function  Range of a Function  Function Notation  Difference Quotient

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition of a Function KFUPM - Prep Year Math Program (c) 2009 All Right Reserved In what follows A and B will denote two nonempty sets. A function from set A to set B is a rule of correspondence that assigns (or maps) to each element x in set A exactly one element y in set B. The set A is called the domain of the function and the subset of the set B consisting of all the outcomes of the function is called the range of the function. The variables x of A and y of B are called the independent variable and dependent variable, respectively.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 1

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2 To do this we perform the following steps must satisfy must satisf

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 3

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Function Notations KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 4

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 5

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Difference Quotients KFUPM - Prep Year Math Program (c) 2009 All Right Reserved The expressions or are called the difference quotients of the function f. These differences have important geometric meaning in relation to the “derivative” of the function f at the point a. In the following examples concentrate on the algebraic techniques used in calculating these quantities.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 6

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 7 a)Does not represent function since the element g has no assignment. b)Does not represent a function since there are two outputs for the input g.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 8 Both (c) and (d) does represent functions from set A to B. Note that, in (d) both the inputs b and g are assigned to the same output u. There is nothing in the definition of a function that prohibits two (or more) inputs to be assigned to the same output.