Wednesday August 24, 2016

Transcription

1 1.1 Functions Wednesday August 24, 2016 EQs: 1. How to write a relation using set-builder & interval notations? 2. How to identify a function from a relation? 1. Subsets of Real Numbers 2. Set-builder Notation Example: Describe the set of numbers using set-builder notation

2 Wednesday August 24, Functions 3. Interval Notations Example: Write each set of numbers using interval notation 4. Relations versus Functions 5. Vertical Line Test

3 1.1 Functions Thursday August 25, 2016 EQs: 1. How to evaluate a function? 2. What is a piecewise function? 6. Find Function Values Example: If g(x) = x2 + 8x 24, find each function value

4 1.1 Functions Thursday August 25, Piecewise Function Example: The average maximum height of children in inches can be modeled as a piecewise function of their parents heights: Find the children s height with given parents height If the parents height is 67in, then the child s height could be 69in If the parents height is 72in, then the child s height could be 78in

5 Monday August 29, Analyzing Graphs of Functions and Relations 1. Find Domain and Range:

6 Monday August 29, Analyzing Graphs of Functions and Relations 1. Find Domain and Range: D = [-8, -4) U (-4, ) D = { x -8 x, x -4, x є R } R = [-10, ) R = { y y -10, y є R } D = [-2, 6) D = { x -2 x < 6, x є R } R = [0, 4] R = { y 0 y 4, y є R } D = (-4, 2) U (2, + ) D = { x x > -4, x 2, x є R } R = [6] U (-2, - ) R = { y y = 6, y < -2, y є R } D = (-6, -2] U [2, + ) D = { x -6 < x -2 or x 2 x є R } R = [6, - ) R = { y y 6, y є R }

7 Monday August 29, Analyzing Graphs of Functions and Relations There are 2 main reasons why domains are restricted: 1. can t divide by 0 2. can t take square (or any even) root of a negative number.

8 Monday August 29, Analyzing Graphs of Functions and Relations 2. Finding intercepts: Intercepts are points where a graph intersect either the x-axis or y-axis. y-intercepts occurs when x=0 x-intercepts occurs when y=0 To find y-intercept of g(x) = x-5-1, we find g(0) g(0) = = 4 Therefore y-intercept is 4 The graph of a function can have 1 or more x-intercepts but only has 1 y-intercept To find x-intercept of f(x) = 2x 2 + x -15, we find the zero(s), root(s), or solution(n) 2x 2 + x -15 = 0 (2x - 5) (x + 3) = 0 (2x - 5) = 0 or (x + 3) = 0 x = 2.5 or x = -3 Therefore the zeros/roots/solutions are -3 and 2.5

9 3. Symmetry of Graphs Monday August 29, Analyzing Graphs of Functions and Relations Example: Test the equation x y 2 = 1 for symmetry Solution: The graph appears to be symmetric with respect to the x-axis. To confirm algebraically, let s replace y with y: x (-y) 2, which is equal to x y 2 To confirm numerically, let s look at the table: Therefore, the equation is symmetric with respect to the x-axis

10 4. Even and Odd Functions Tuesday August 30, Analyzing Graphs of Functions and Relations Example 1: Is the function f(x) = x 3-2x odd, even, or neither? Solution: the function f(x) = x 3-2x is odd because f(-x) = -f(x) Example 2: Is the function g(x) = x odd, even, or neither? Solution: the function g(x) = x is even because g(-x) = g(x) Example 2: Is the function h(x) = x 3 0.5x 2 3x odd, even, or neither? Solution: the function h(x) = x 3 0.5x 2 3x is neither Because h(-x) h(x) and h(-x) -h(x)

11 Wednesday August 31, Continuity, End Behavior, and Limits 1. Definition of Limits: If the value of f(x) approaches a unique value L as x approaches c from each side, then the limit of f(x) as x approaches c is L 2. Continuity: Continuous Function has no breaks, holes, or gaps Discontinuous Function is not continuous Types of Discontinuity: A function has an infinite discontinuity at x =c if the function value (y) increases or decreases indefinitely as x approaches c from the left and right A function has a jump discontinuity at x =c if the limits of the function as x approaches c from the left and right exist but has 2 distinct values A function has a removable discontinuity if the function is continuous everywhere except for a whole at x = c Continuity Test: A function is continuous at x = c if 1. f(c) exists 2. f(x) approaches the same value from either side of c 3. The value that f(x) approaches from each side of c is f(c)

12 Wednesday August 31, Continuity, End Behavior, and Limits Determine if a function is continuous using the continuity test: Example: Solution: Step 1: Does f(-3) exist? Yes, f(-3) = 2 (-3) = 5 Step 2: Investigate f(x) when x is approaching -3 As x approaches -3 from the left, f(x) approaches 5 As x approaches -3 from the right f(x) approaches -11 Therefore, f(x) is not continuous and it s a jump discontinuity Example: Solution: Step 1: f(-3) = 0/0 and f(3) = 6/0, so f(-3) and f(3) do not exist. Therefore f(x) is discontinuous at -3 and 3 Step 2: when x is approaching -3, f(x) approaches but when x is approaching 3 from the left, f(x) approaches large negative number. when x is approaching 3 from the right, f(x) approaches large positive number. Therefore, f(x) is not an infinite discontinuity at x=3

13 Thursday September 1, Continuity, End Behavior, and Limits 3. End Behavior: how a function behaves at either end of the graph. The behavior is determined by the degree and leading coefficient sign. To predict: 1. Write in general form (highest to lowest degree) 2. Determine if the highest degree is odd or even 3. Determine if the coefficient is positive or negative Example: Describe the end behavior of f(x) = -x 4 + 8x 3 + 3x 2 + 6x -80 Solution:

14 1.4 Extrema and Average Rates of Changes 1. Increasing/Decreasing Behavior: Thursday September 1, 2016 Critical points: points where the slope is zero Extrema: points where a function changes its increasing or decreasing behavior Max/Min value: either relative or absolute Point of Inflection: point where a graph changes its shape

15 Thursday September 1, Extrema and Average Rates of Changes Example: Estimate and classify the extrema for the following graph Relative max at x = -.5 Relative min at x =1 No absolute extrema Rel max at x =-1.5 Rel min at x = -.3 Absolute max at x = 1 Rel max at x =-1 and 0.7 Rel min at x = 0 and 2 No absolute max Average Rate of Change:

16 Thursday September 1, Extrema and Average Rates of Changes Example: Find the average rate of change of f(x) = - x 3 + 3x on the following interval a. [-2, -1] b. [0,1]