Concept of a Function. Brian Benson Math 100

Transcription

1 Concept of a Function

2 Concept of a Function Definition of a Function

3 Concept of a Function Definition of a Function Vertical Line Test

4 Concept of a Function Definition of a Function Vertical Line Test Domain and Range

5 Concept of a Function Definition of a Function Vertical Line Test Domain and Range Symmetry (of inverses)

6 Concept of a Function Definition of a Function Vertical Line Test Domain and Range Symmetry (of inverses) Shifting and Scaling of Parent Functions

7 Concept of a Function Definition of a Function Vertical Line Test Domain and Range Symmetry (of inverses) Shifting and Scaling of Parent Functions One-to-one Functions and Inverses Horizontal Line Test

8 Types of Functions Linear

9 Types of Functions Linear Quadratics

10 Types of Functions Linear Quadratics Higher Degree Polynomials

11 Types of Functions Linear Quadratics Higher Degree Polynomials Rational

12 Types of Functions Linear Quadratics Higher Degree Polynomials Rational Exponential and Logarithmic

13 Types of Functions Linear Quadratics Higher Degree Polynomials Rational Exponential and Logarithmic Root

14 Types of Functions Linear Quadratics Higher Degree Polynomials Rational Exponential and Logarithmic Root Absolute Value

15 Types of Functions Linear Quadratics Higher Degree Polynomials Rational Exponential and Logarithmic Root Absolute Value

16 Types of Functions Linear Quadratics Higher Degree Polynomials Rational Exponential and Logarithmic Root Absolute Value We considered equations with equality and/or inequalities involving each of these functions. Often these were in the form of word problems where you needed to set up your own equations (see homework). We also considered systems of linear equations.

17 iclicker Question 1 Solve and CHECK your work: x = x + 2

18 iclicker Question 1 Solve and CHECK your work: x = x + 2 A. x = 2 B. x = 1 C. Both A. and B. D. x = 3 E. None of the above

19 Important Formulas for Linear Functions

20 Important Formulas for Linear Functions Slope m of a line connecting two points (x 1, y 1 ) and (x 2, y 2 ): m = y 2 y 1 x 2 x 1

21 Important Formulas for Linear Functions Slope m of a line connecting two points (x 1, y 1 ) and (x 2, y 2 ): m = y 2 y 1 x 2 x 1 Slope-intercept form of a line: y = mx + b (m: slope, b: y-intercept)

22 Important Formulas for Linear Functions Slope m of a line connecting two points (x 1, y 1 ) and (x 2, y 2 ): m = y 2 y 1 x 2 x 1 Slope-intercept form of a line: y = mx + b (m: slope, b: y-intercept) Point-slope form of a line: y y 0 = m(x x 0 ) (where (x 0, y 0 ) is a point on the line)

23 Important Formulas for Quadratics

24 Important Formulas for Quadratics General form of a parabola: y = ax 2 + bx + c

25 Important Formulas for Quadratics General form of a parabola: y = ax 2 + bx + c Vertex form of a parabola: y = a(x h) 2 + k

26 Important Formulas for Quadratics General form of a parabola: y = ax 2 + bx + c Vertex form of a parabola: y = a(x h) 2 + k Quadratic formula, which gives solutions to the equation ax 2 + bx + c = 0, a 0 x = b ± b 2 4ac 2a

27 Important Formulas for Quadratics General form of a parabola: y = ax 2 + bx + c Vertex form of a parabola: y = a(x h) 2 + k Quadratic formula, which gives solutions to the equation ax 2 + bx + c = 0, a 0 x = b ± b 2 4ac 2a Vertex formula for vertex (h, k) of ax 2 + bx + c (found from completing square): h = b 2a, k = function value at h

28 Formulas for Compound Interest

29 Formulas for Compound Interest FV : Future value

30 Formulas for Compound Interest FV : Future value PV : Present value

31 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest

32 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest n: number of times compounded per year

33 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest n: number of times compounded per year t: number of years

34 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest n: number of times compounded per year t: number of years For a finite number of times compounded: ( FV = PV 1 + r ) nt. n

35 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest n: number of times compounded per year t: number of years For a finite number of times compounded: ( FV = PV 1 + r ) nt. n Continuously compounded: ( FV = PV (e rt ) = lim (1 PV + r ) ) nt n n

36 Formulas for Compound Interest FV : Future value PV : Present value r: annual interest n: number of times compounded per year t: number of years For a finite number of times compounded: ( FV = PV 1 + r ) nt. n Continuously compounded: ( FV = PV (e rt ) = lim (1 PV + r ) ) nt n n Want to understand the right hand side of this equation? TAKE CALCULUS - MATH 220!!!

37 Logarithm Properties

38 Logarithm Properties For x, y > 0, b > 0 with b 1, and n a real number: 1. log b (xy) = log b (x) + log b (y)

39 Logarithm Properties For x, y > 0, b > 0 with b 1, and n a real number: 1. log b (xy) = log b (x) + log b (y) ( ) 2. log x b y = log b (x) log b (y)

40 Logarithm Properties For x, y > 0, b > 0 with b 1, and n a real number: 1. log b (xy) = log b (x) + log b (y) ( ) 2. log x b y = log b (x) log b (y) 3. log b (x n ) = n log b (x)

41 Rational Functions

42 Rational Functions Finding horizontal, vertical, and slant asymptotes

43 Rational Functions Finding horizontal, vertical, and slant asymptotes Finding zeros and y-intercept

44 Rational Functions Finding horizontal, vertical, and slant asymptotes Finding zeros and y-intercept Finding poles

45 iclicker Question 2 What are the zeros of the following rational function? r(x) = x 2 5x + 4 x 2 + 4x + 3

46 iclicker Question 2 What are the zeros of the following rational function? r(x) = x 2 5x + 4 x 2 + 4x + 3 A. x = 4 B. x = 1 C. x = 1 D. All of the above E. Both A. and B.

47 System of Equations To solve systems of equations we can use:

48 System of Equations To solve systems of equations we can use: Substitution

49 System of Equations To solve systems of equations we can use: Substitution Elimination

50 System of Equations To solve systems of equations we can use: Substitution Elimination Graphing

51 System of Equations To solve systems of equations we can use: Substitution Elimination Graphing Matrix row operations

52 System of Equations To solve systems of equations we can use: Substitution Elimination Graphing Matrix row operations Matrix inverse

53 System of Equations To solve systems of equations we can use: Substitution Elimination Graphing Matrix row operations Matrix inverse

54 Example

55 Example Solve for s: ( s ) ln =

56 Example

57 Example The number of gizmos demanded each year is given by the formula D(x) = log(x + 3) where D(x) is in thousands, x is the number of years since 1980, and x > 0. In what year were gizmos demanded?

58 Example

59 Example What is the domain of f (x) = ln( 7x + 9)?

60 Example Find all real numbers x so that 2x + 3 = 5x 6.

61 Example Find all real numbers x so that 5x + 3 > 9.

62 Example If f (x) = 4x 1 3, then find f 1 (x).

63 Example Solve the quadratic inequality x 2 + (π e)x πe < 0.

64 iclicker Question 3 Which of the following is a polynomial with a single root at x = 4 and a double root at x = 7?

65 iclicker Question 3 Which of the following is a polynomial with a single root at x = 4 and a double root at x = 7? A. (x 4)(x 7) 2 B. (x 4)(x 2 7) C. (x + 4)(x 2 + 7) D. (x + 4)(x + 7) 2 E. None of the above