What is a Function? How to find the domain of a function (algebraically) Domain hiccups happen in 2 major cases (rational functions and radicals)

Transcription

1 What is a Function? Proving a Function Vertical Line Test Mapping Provide definition for function Provide sketch/rule for vertical line test Provide sketch/rule for mapping (notes #-3) How to find the domain of a function (algebraicall) hiccups happen in major cases (rational functions and radicals) Rational functions Rule Denominator cannot equal 0 What to do Eample Set denominator equal to 0 and solve for f() = (, ) (, ) Radicals Radicand must be positive (greater than or equal to 0). Set radicand greater than or equal to 0 and solve for g() = [5, ) Even Odd Neither (, ) = (, ) (, ) = (, ) Provide definition for graphicall Graphicall Smetric about Graphicall Smetric about Graphicall Provide definition for algebraicall (notes #-5) Algebraicall* Algebraicall* Algebraicall*

2 Function Behavior Tpes of Continuit *alwas view the function left to right Write intervals in terms of Increasing The and values increase Decreasing The values increase, the values decrease Constant The values incrase, the values remain constant Decreasing (, 0] Constant ( 7, ] Increasing (0, ) Definition Picture Continious for all of Removeable Discontunit Tpes of Continuit Definition for each tpe Picture of each tpe Notes #-4 Definition Picture Jump Discontunit Definition for each tpe Picture of each tpe Notes #-4 Infinite Discontunit

3 Horizontal Asmptotes f() = a m m a n n Case : Case : Case 3: If the same degree is in the top and bottom; ratio of the coefficients of largest term Larger eponent is in the denominator, = 0 Larger eponent is in the numerator, no HA (it has a slant asmptote) m = n m < n m > n f() = ; = am a n = 3 f() = + 3 ; = 0 f() = No HA! Vertical Asmptotes Case : Case : Case 3: Rule for each case Sketch for each case Equation eample for each case Notes #-5

4 LINEAR Continuit: Asmptotes: f() = Continuous Odd none Quadratic f() = Continuit: Asmptotes: Continuous Even None Cubic f() = 3 Continuit: Pg. 76 Asmptotes:

5 = VA Reciprocal* = HA Continuit: Asmptotes: f() = This gu is a beast he has an additional card located in chapter material. Ma require snthetic division! Pg. 9 [0, 6] 0 4 Square Root f() = Continuit : Asmptot es: Pg Eponential* f() = e Continuit: Asmptotes: There is an eponential function we use for transformations see pg. Pg. 56

6 Absolute Value f() = Continuit: Asmptotes: Continuous Even none [0, 4] 0 Natural Log f() = ln Continuit: Asmptote s: Pg Logistic Function f() = Continuit: Asmptotes: + e Pg. 59

7 [0, π] 0 0 π π 0 3π π 0 Sine f() = sin Continuit: Asmptotes: [0, π] 0 0 π π 0 3π π 0 Cosine f() = cos Continuit: Asmptotes: Greatest Integer 0 0 f() = int () (, ) All integers Continuit: Jump discontinuit Nonremoveable none Asmptotes: none Integers are numbers without decimals (positive & negative whole numbers). Alwas defaults to smaller integer.

8 Eponential Function for Transformations (chpt. 3) [, ] f() = b Continuit: Asmptote: Log Function for Transformations (chpt. 3) [ b, b] f() = log b Continuit: Asmptote: a b c d Transformations f() = a (b c) + d What does it do? a > 0: a < 0: a < 0 (a is negative): b > 0 b < 0: b < 0 (b is negative): c > 0 (positive): c < 0 (negative): d > 0 (positive): d < 0 (negative): Appling Transformation Affects & Vertical Asmp Affects & Horizontal Asmp

9 Contents Page Topic Page Topic What is a function 3 Square Root Finding 4 Eponential 3 Smmetr 5 Absolute Value 4 Function Behavior 6 Natural Log 5-6 Tpes of Discontinuities 7 Logistic 7 Horizontal Asmptotes 8 Sine 8 Vertical Asmptotes 9 Cosine 9 Linear 0 Greatest Integer 0 Quadratic Eponential transformations Cubic Log - transformations Reciprocal 3 Transformations M Precalculus FUNCTION LIBRARY Name Pd