Chapter 2. Functions and Graphs. 03 Feb 2009 MATH 1314 College Algebra Ch.2 1

Transcription

1 Capter Functions and Graps 03 Feb 009 MATH 1314 College Algebra C. 1

2 .1 Basics of Functions & Teir Graps 03 Feb 009 MATH 1314 College Algebra C.

3 Objectives Find te domain & range of a relation. Evaluate a function. Grap functions by plotting points. Obtain information from a grap. Identify te domain & range from a grap. Identify -y intercepts from a grap. 03 Feb 009 MATH 1314 College Algebra C. 3

4 Domain & Range Domain: first components in te relation (independent variable or -values) Range: second components in te relation (dependent variable, te value depends on wat te domain value is, aka y-values) Functions are SPECIAL relations: A domain element corresponds to eactly ONE range element. 03 Feb 009 MATH 1314 College Algebra C. 4

5 EXAMPLE Consider te function: eye color (Assume all people ave only one color) It IS a function because wen asked te eye color of eac person, tere is only one answer. e.g. {(Joe, brown), (Mo, blue), (Mary, green), (Ava, brown), (Natalie, blue)} NOTE: te range values are not necessarily unique. 03 Feb 009 MATH 1314 College Algebra C. 5

6 Evaluating a function Common notation: f() = function Evaluate te function at various values of, represented as: f(a), f(b), etc. Eample: f() = 3 7 f() = 3() 7 = 6 7 = -1 f(3 ) = 3(3 ) 7 = = 3 03 Feb 009 MATH 1314 College Algebra C. 6

7 Graping a functions Horizontal ais: values Vertical ais: y values Plot points individually or use a graping utility (calculator or computer algebra system) Eample: y = 1 03 Feb 009 MATH 1314 College Algebra C. 7

8 Table of function values y = 1 X (domain) Y (range) Feb 009 MATH 1314 College Algebra C. 8

9 Graps of functions 03 Feb 009 MATH 1314 College Algebra C. 9

10 Can you identify domain & range from te grap? Look orizontally. Wat all -values are contained in te grap? Tat s your domain! Look vertically. Wat all y-values are contained in te grap? Tat s your range! 03 Feb 009 MATH 1314 College Algebra C. 10

11 Wat is te domain & range of te function wit tis grap? 1) Domain:(, ), Range:(, ) ) Domain:( 3, ), Range:(, ) 3) Domain:( 3, ), Range(: 3, ) 4) Domain:(, ), Range:( 3, ) Correct Answer: 4 03 Feb 009 MATH 1314 College Algebra C. 11

12 Finding intercepts: X-intercept: were te function crosses te -ais. Wat is true of every point on te -ais? Te y-value is ALWAYS zero. Y-intercept: were te function crosses te y-ais. Wat is true of every point on te y-ais? Te -value is ALWAYS zero. Can te -intercept and te y-intercept ever be te same point? YES, if te function crosses troug te origin! 03 Feb 009 MATH 1314 College Algebra C. 1

13 . More of Functions and Teir Graps 03 Feb 009 MATH 1314 College Algebra C. 13

14 Objectives Find & simplify a function s difference quotient. Understand & use piecewise functions. Identify intervals on wic a function increases, decreases, or is constant. Use graps to locate relative maima or minima. Identify even or odd functions & recognize te symmetries. 03 Feb 009 MATH 1314 College Algebra C. 14

15 Difference Quotient Useful in discussing te rate of cange of function over a period of time EXTREMELY important in calculus, ( represents te difference in two values) f ( ) f ( ) 03 Feb 009 MATH 1314 College Algebra C. 15

16 03 Feb 009 MATH 1314 College Algebra C. 16 Find te difference quotient 6 6 ) ( ) ( ) 6 (6 6 6 ) ( ) ( 1) ( ) ( ) ( ) ( 1 ) 3 3 ( ) ( 1 ) ( ) ( ) ( 1 ) ( = = = = = = = = f f f f f f f f f f

17 Wat is a piecewise function? A function tat is defined differently for different parts of te domain. Eamples: You are paid $10/r for work up to 40 rs/wk and ten time and a alf for overtime. f( ) 10 if 40 = 15 if > Feb 009 MATH 1314 College Algebra C. 17

18 Increasing and Decreasing Functions Increasing: Grap goes up as you move from left to rigt. 1 <, f ( 1 ) < f ( ) Decreasing: Grap goes down as you move from left to rigt. < f ( ) > f ( ) 1, 1 Constant: Grap remains orizontal as you move from left to rigt. 1 <, f ( 1 ) = f ( ) 03 Feb 009 MATH 1314 College Algebra C. 18

19 Even & Odd Functions Even functions are tose tat are mirrored troug te y-ais. (if replaces, te y value remains te same) (e.g. 1 st quadrant reflects into te nd quadrant) Odd functions are tose tat are rotated troug te origin. (if replaces, te y value becomes y) (e.g. 1 st quadrant reflects into te 3 rd quadrant) 03 Feb 009 MATH 1314 College Algebra C. 19

20 Determine if te function is even, odd, or neiter. f ( ) = ( 4) 1. Even. Odd 3. Neiter Correct Answer: 3 03 Feb 009 MATH 1314 College Algebra C. 0

21 .3 Linear Functions & Slope 03 Feb 009 MATH 1314 College Algebra C. 1

22 Objectives Calculate a line s slope. Write point-slope form of a line s equation. Model data wit linear functions and predict. 03 Feb 009 MATH 1314 College Algebra C.

23 Wat is slope? Te steepness of te grap, te rate at wic te y values are canging in relation to te canges in. How do we calculate it? slope = m = Δy Δ = y y Feb 009 MATH 1314 College Algebra C. 3

24 A line as one slope Between any pts. on te line, te slope MUST be te same. Use tis to develop te point-slope form of te equation of te line. y y = m( 1) 1 Now, you can develop te equation of any line if you know eiter a) points on te line or b) one point and te slope. 03 Feb 009 MATH 1314 College Algebra C. 4

25 03 Feb 009 MATH 1314 College Algebra C. 5 Find te equation of te line tat goes troug (,5) and (-3,4) 1 st : Find slope of te line m= nd : Use eiter point to find te equation of te line & solve for y ) ( 4 5 = ) ( = = = y y

26 .5 Transformation of Functions Recognize graps of common functions Use vertical sifts to grap functions Use orizontal sifts to grap functions Use reflections to grap functions Grap functions w/ sequence of transformations 03 Feb 009 MATH 1314 College Algebra C. 6

27 Vertical sifts Moves te grap up or down Impacts only te y values of te function No canges are made to te values Horizontal sifts Moves te grap left or rigt Impacts only te values of te function No canges are made to te y values 03 Feb 009 MATH 1314 College Algebra C. 7

28 Recognizing te sift from te equation. Eamples of sifting te function f() = Vertical sift of 3 units up f ( ) =, ( ) = 3 Horizontal sift of 3 units left (HINT: s go te opposite direction tat you migt believe.) f ( ) =, g( ) = ( 3) 03 Feb 009 MATH 1314 College Algebra C. 8

29 Combining a vertical & orizontal Eample of function tat is sifted down 4 units and rigt 6 units from te original function. sift f ( ) =, g( ) = Feb 009 MATH 1314 College Algebra C. 9

30 Reflecting Across -ais (y becomes negative, -f()) Across y-ais ( becomes negative, f(-)) 03 Feb 009 MATH 1314 College Algebra C. 30

31 .6 Combinations of Functions; Composite Functions Objectives Find te domain of a function Form composite functions. Determine domains for composite functions. Write functions as compositions. 03 Feb 009 MATH 1314 College Algebra C. 31

32 Using basic algebraic functions, wat limitations are tere wen working wit real numbers? A) You can never divide by zero. Any values tat would result in a zero denominator are NEVER allowed, terefore te domain of te function (possible values) would be limited. B) You cannot take te square root (or any even root) of a negative number. Any values tat would result in negatives under an even radical (suc as square roots) result in a domain restriction. 03 Feb 009 MATH 1314 College Algebra C. 3

33 Eample Find te domain 5 6 Tere are s under an even radical AND s in te denominator, so we must consider bot of tese as possible limitations to our domain. 5 ( 3)( Domain 0, 6 0 ) : { :,3 3} 03 Feb 009 MATH 1314 College Algebra C. 33 0, >,

34 Composition of functions Composition of functions means te output from te inner function becomes te input of te outer function. f(g(3)) means you evaluate function g at =3, ten plug tat value into function f in place of te. Notation for composition: ( f o g)( ) = f ( g( )) 03 Feb 009 MATH 1314 College Algebra C. 34

35 .7 Inverse Functions Objectives Verify inverse functions Find te inverse of a function. Use te orizontal line test to deterimine oneto-one. Given a grap, grap te inverse. Find te inverse of a function & grap bot functions simultaneously. 03 Feb 009 MATH 1314 College Algebra C. 35

36 Wat is an inverse function? A function tat undoes te original function. A function wraps an and te inverse would unwrap te resulting in wen te functions are composed on eac oter. f ( f 1 ( )) = f 1 ( f ( )) = 03 Feb 009 MATH 1314 College Algebra C. 36

37 How do teir graps compare? Te grap of a function and its inverse always mirror eac oter troug te line y=. Eample:y = (1/3) and its inverse = 3(-) Every point on te grap (,y) eists on te inverse as (y,) (i.e. if (-6,0) is on te grap, (0,-6) is on its inverse. 03 Feb 009 MATH 1314 College Algebra C. 37

38 Do all functions ave inverses? Yes, and no. Yes, tey all will ave inverses, BUT we are only interested in te inverses if tey ARE A FUNCTION. DO ALL FUNCTIONS HAVE INVERSES THAT ARE FUNCTIONS? NO. Recall, functions must pass te vertical line test wen graped. If te inverse is to pass te vertical line test, te original function must pass te HORIZONTAL line test (be one-to-one)! 03 Feb 009 MATH 1314 College Algebra C. 38

39 How do you find an inverse? Undo te function. Replace te wit y and solve for y. 03 Feb 009 MATH 1314 College Algebra C. 39