Function? c. {(-1,4);(0,-4);(1,-3);(-1,5);(2,-5)} {(-2,3);(-1,3);(0,1);(1,-3);(2,-5)} a. Domain Range Domain Range

Transcription

1 Section 3.1: Functions Definitions (pages ): A relation is a correspondence between two sets. A function is a correspondence to a first set, called the domain, to a second set, called the range, such that each member of the domain corresponds to exactly one (a unique) member of the range. Examples: Which of each pair of relations represent functions? a. Domain Range Domain Range b. Domain Range Domain Range People Phone Numbers People Birth Dates c. {(-1,4);(0,-4);(1,-3);(-1,5);(2,-5)} {(-2,3);(-1,3);(0,1);(1,-3);(2,-5)} 1

2 Section 3.1: Functions Example: y = x 2 x = y 2 The graphs illustrate the Vertical Line Test for Functions. See page

3 Section 3.1: Functions Function Notation The two variable equation y= x in function x+ 1 notation is written f ( x) = x. Variable y is replaced x + 1 by f ( x ) read as f of x or f evaluated at x. Input Function Output Equation x f f ( x ) f ( x) = x x + 1 Independent Mapping Dependent Rule Variable Variable Refer to f ( x ) as the value of f at the number x; f is the name of the function. For function y= f ( x), input variable x is the independent variable; output y is the dependent variable (other variable names can be used). The variable x is also called the argument of the function. Evaluating Functions: If f ( x) = x (note: f ( ) = ), x Examples: Find f ( 2), f ( 1), f ( a ), f ( x) vs. f ( x), f 1 vs. f (1), and f ( x ) + 2 vs. f ( x+ 2) vs. f ( x) + f (2). 2 f (2) 3

4 Section 3.1: Functions A difference quotient is a fundamental idea in calculus. Evaluating a difference quotient f ( x+ h) f ( x) : h Page Definition: The domain of a function is the largest set of real numbers for which the value f ( x ) is defined. Examples: Find the domains for the functions and express in interval notation: a. f x x x x 4 3 ( ) g( x) = = + + b. 2 x 2x 6 3x 10 c. h( x) = 2x 6 a. The function f ( x ) illustrates that any polynomial has the set of all real numbers (, ) as its domain. b. The function g( x ) illustrates that because division by zero is undefined, rational function domains will have some excluded values. c. The function h( x ) illustrates that because negative values for radicands of square roots result in complex numbers, domains will have restricted intervals. Compare this with examples 10b and 10c on page Example: Page

5 Section 3.2: Graphs of Functions Reminder: A function is a correspondence between a first set, the domain, to a second set, the range, so that each member of the domain corresponds to exactly one (a unique) member of the range. Examples: y = x 2 x = y 2 Example: y = x 3 y = 1/ x 5 Exam 1 on Friday 2/11/11 will cover: Sections 1.3, 1.6, , and 3.1. A review guide is online.

6 Section 3.2: Graphs of Functions Everyone has College Algebra Basic Function Graphs. Discuss and label the symmetry of the graphs, even vs. odd. Function Symmetric w.r.t. Even y-axis Odd origin Even/Odd Examples: Page , 22, 18 Increasing, Decreasing, and Constant Functions (page 249) - Always report an open x-interval. See Example on page 249. Example: Page 258, #26 f ( x ) f ( x ) Average Rate of Change = 2 1 x x (See diagram on page 251.) Evaluating a difference quotient (See diagram on page 253.) Page 259: 44, 92b 2 1 f ( x+ h) f ( x). h 6

7 Section 3.2: Graphs of Functions Piecewise-Defined Graphs Page x + 1 if 3 x < 0 f ( x) = 4 if x = 0 x 2 if x > 0 Domain: Range: Increasing Interval(s): Increasing Interval(s): Constant Interval(s) : On Thursday, everyone will turn in College Algebra Basic Function Graphs fully completed, including the domains and ranges. Make certain you are neat and precise. Your numbers should be written in integer or fractional form. 7

8 Section 3.3 Graphing Techniques: Transformations I. Rigid Translations Example: For f ( x) = x, graph y= f ( x), y= f ( x) + 2, y= f ( x) 3. For any function y= f ( x) and c> 0, y= f ( x) + c is a vertical shift upward and y= f ( x) c is a vertical shift downward, page 265. Example: For f ( x) = x, graph y= f ( x), y= f ( x+ 2), y= f ( x 3). For any function y= f ( x) and c> 0, the function y= f ( x+ c) is a horizontal shift left, and y= f ( x c) is a horizontal shift right, page 265. Example: For f ( x) = x, graph y= f ( x), y= f ( x), and y= f ( x). For any function y= f ( x), the function y= f ( x) is a reflection about the x-axis of the graph of y= f ( x) and y= f ( x) is a reflection about the y-axis, page

9 Section 3.3 Graphing Techniques: Transformations II. Non-Rigid Transformations Example: For f ( x) = x2, graph y= f ( x), y= 1/2 f ( x), and y= 2 f ( x). For any function y= f ( x), the new function y= cf ( x) is a vertical shrink (compression) if 0< c< 1and a vertical stretch if c> 1, page 271. Graph by multiplying the basic function s y-values by c. Example: For f ( x) = x2, graph y= f ( x), y= f (1/2 x), and y= f (2 x). For any function y= f ( x), the new function y= f ( cx) is a horizontal shrink (compression) if c> 1 and a horizontal stretch if 0< c< 1, page 272. Graph by dividing the basic function s x-values by c. Summary of Transformations of y= f ( x), page 274 Examples Page 274 Mention 1-12, 17, 18, 21, 26, 32, 48, 66 9

10 Section 3.4: The Algebra of Functions I. Arithmetic Operations on Functions Two functions f and g with overlapping domains can be combined arithmetically: 1. Sum: ( f + g)( x) = f ( x) + g( x) 2. Difference: ( f g)( x) = f ( x) g( x) 3. Product: ( f g)( x) = f ( x) g( x) 4. Quotient: ( f / g)( x) = f ( x)/ g( x), g( x) 0 The domains of all functions are the intersection of the domains of f and g, {domain f} {domain g}, except for f / g the zeros of the denominator must be omitted. Page 286 2, 10 II. Function Composition The composition of the function f with the function g is defined as ( f g)( x) = f ( g( x)). The domain of f g is the set of all x in the domain of g such that g( x ) is in the domain of f. To find the domain, find the domain of ( f g)( x) = f ( g( x)) and then exclude x values that are not in the domain of g( x ). Page , 20 (mention), 26, 38, 50, 60, 64 10

11 Section 3.5: One-to-one Functions & Inverse Functions Reminders: A relation is a correspondence between two sets. An inverse relation is produced by interchanging the first and second coordinates of ordered pairs or by interchanging the variables of an x-y equation. A function is a relation such that each domain value is assigned exactly one range value. The Vertical Line Test states if any vertical line intersects the graph of a relation at more than one point, the relation is not a function. Definition: A function is one-to-one (1 1) if for any choice of domain values a and b witha b, then f ( a) f ( b), or if f ( a) = f ( b), then a= b, page 290. Example: Domain Range Domain Range Domain Range Relation? Relation? Relation? One-to-One? One-to-One? One-to-One? 11

12 Section 3.5: One-to-one Functions & Inverse Functions Example: {(-2,1),(-1,2),(1,4),(-1,5)} {(-2,1),(-1,1),(1,3),(2,4)} {(-2,1),(-1,2),(1,4),(2,5)} One-to-One? One-to-One? One-to-One? Inverse Relation? Inverse Relation? Example: x= y 2 y= x2 y= x2, x 0 One-to-One? One-to-One? One-to-One? Horizontal Line Test: If any horizontal line intersects the graph of a function at more than one point, then the function is not 1 1 (page 290). A strictly increasing (or decreasing) function is 1 1. One-to-one functions are important because all one-toone functions have inverse functions. 12

13 Section 3.5: One-to-one Functions & Inverse Functions Definition : If y= f ( x) is a one-to-one function with domain A and range B, the inverse function of f, denoted by f 1 has domain B and range A and is defined by b= f ( a) a= f 1( b) for any b in B. Inverse Function Property (page 293): If y= f ( x) a nd y= f 1( x) are inverse functions, then ( f 1 f )( x) = f 1( f ( x)) = x for each x in domain f and ( f f 1)( x) = f ( f 1( x)) = x for each x in domain f 1. Example: Verify that f ( x) = x3and f 1( x) = 3x are inverse functions. Compare the table values of these functions to see that these facts hold for inverse functions: Domain of f = Range of f 1 Range of f = Domain of f 1 Compare the graphs of f ( x) = x3 and f 1( x) = x1/3. Notice (using ZoomSquare on your calculator) that they are symmetric with respect to the line y= x. 13

14 Section 3.5: One-to-one Functions & Inverse Functions The graphs of inverse functions y= f ( x) and y= f 1( x) are symmetric with respect to the line y = x (page 294). Of the functions studied so far, the horizontal line test indicates that many are not one-to-one and, so, do not have inverses. For those that do, use the following procedure to find them: Obtaining an Inverse Function Formula (page 296) Page , 54, 58, 62, 64 14