1. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 1.2 Basics of Functions and Their Graphs

2. Blitzer, College Algebra: An Early Functions Approach, 2ed 22 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Relations The test grades of 22 students in Mrs. Smith’s Algebra class are shown in the table below. The table indicates a correspondence between a grade and the number of students receiving that grade. Grade# of Students A5 B7 C8 D1 F1 We can write this correspondence as a set of ordered pairs: {(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}. The mathematical term for a set of ordered pairs is a relation.

3. Blitzer, College Algebra: An Early Functions Approach, 2ed 33 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Relations Example: Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation. Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation. Find the domain and range of the relation: {(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}. Domain: {A, B, C, D, F}Range: {5, 7, 8, 1} The 1 is only listed once.

4. Blitzer, College Algebra: An Early Functions Approach, 2ed 44 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Functions Grade# of Students A5 B7 C8 D1 F1 Grade ABCDFABCDF # of Students 57815781 Domain Grades corresponding to number of students. Range A relation in which each member of the domain corresponds to exactly one member of the range is a function. This relation is a function.

5. Blitzer, College Algebra: An Early Functions Approach, 2ed 55 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Functions Grade# of Students A5 B7 C8 D1 F1 57815781 Grade ABCDFABCDF Domain Number of students corresponding to grades. Range Definition of a Function A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. Definition of a Function A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. This relation is NOT a function.

6. Blitzer, College Algebra: An Early Functions Approach, 2ed 66 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Functions Example: Determine whether the relation is a function: {(3, 2), (0, 8), (2, 3), (3, 4), (4, 4)}. 30243024 28342834 DomainRange This is not a function because 3 corresponds to both 2 and 4. A function can have two different first components with the same second component. By contrast, a relation is not a function when two different ordered pairs have the same first component and different second components

7. Blitzer, College Algebra: An Early Functions Approach, 2ed 77 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Functions as Equations Functions are usually given in terms of equations rather than as sets of ordered pairs. y = 3x + 5 For each value of x, there is only one value of y. y is a function of x. The variable x is called the independent variable because it can be assigned any value from the domain. The variable y is called the dependent variable because its value depends on x.

8. Blitzer, College Algebra: An Early Functions Approach, 2ed 88 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Functions as Equations Example: Determine whether the equation defines y as a function of x. x 2 + y 2 = 9 Solve the equation for y in terms of x x 2 + y 2 = 9 x 2 + y 2 – x 2 = 9 – x 2 Solve for y. y 2 = 9 – x 2 Simplify. Since two or more values of y can be obtained for a given x, the equation is not a function. Apply the square root property.

9. Blitzer, College Algebra: An Early Functions Approach, 2ed 99 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Function Notation If an equation in x and y gives one and only one value of y for each value of x then the variable y is a function of the variable x. The special notation f(x), read “f of x” or “f at x,” represents the value of the function at the number x. x f(x) f(x) = 3x 2 + 8 Input Output Equation This is read as f(x) = 3x 2 + 8.

10. Blitzer, College Algebra: An Early Functions Approach, 2ed 10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Evaluating a Function Example: Evaluate the function for f(– 2). f(x) = 3x 2 + 8 f(x) = 3(– 2) 2 + 8 Substitute – 2 for x in the equation. = 3(4) + 8 Square – 2. = 12+ 8 Multiply. = 12+ 8 Add. = 20 Simplify.

11. Blitzer, College Algebra: An Early Functions Approach, 2ed 11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Graphs of Functions Example: Graph the function f(x) = x + 3. xf(x) = – x + 3(x, y) 1y = (1) + 3 = 4(1, 4) 2y = (2) + 3 = 5(2, 5) 11 y = (– 1) + 3 = 2 (  1, 2) (1, 4) x y 11 22 33 44 123 4 22 33 44 1 2 3 4 (  1, 2) (2, 5) f(x) = – x + 3 The graph of a function is the graph of its ordered pairs.

12. Blitzer, College Algebra: An Early Functions Approach, 2ed 12 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. The Vertical Line Test The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. FUNCTION NOT A FUNCTION FUNCTION