last night’s hw 6. no 8. yes 32a. 2 32b. 5 32c. √x+2 33a. -1/9 33b. undefined 33c....
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Last Night’s HWLast Night’s HW
6. no6. no 8. yes8. yes 32a. 232a. 2 32b. 532b. 5 32c. 32c. √x+2√x+2 33a. -1/933a. -1/9 33b. undefined33b. undefined 33c. 1/(y^2 + 6y)33c. 1/(y^2 + 6y)
66. D: (- ∞, 0) U (0, 2) U (2, ∞)68. D: (-6, ∞)70. D: (- ∞, -3) U (3, ∞)
Essential QuestionEssential Question
How do you determine the relative max, How do you determine the relative max, min, intervals of increase, decrease, even min, intervals of increase, decrease, even and odd functions?and odd functions?
1.2 Graphs of Functions1.2 Graphs of Functions
Example 1: Given the graph shown, answer the Example 1: Given the graph shown, answer the following questions?following questions?
a)a) What is f(0) ?What is f(0) ?b)b) What is f(9)?What is f(9)?c)c) What is the domain?What is the domain?d)d) What is the range?What is the range?e)e) What are the x-intercepts?What are the x-intercepts?f)f) How often does the line y = -1 intercept the How often does the line y = -1 intercept the
graph?graph?
a. f(0)=-3
(-4, 2)
(-2, 0)
(0, -3)
(1, -2)
(3, 0)
(5, 4)
(7, 0)
(9, -2)
(10, -1)
b. f(9)=is undefinedc. D=[-4, 9) U (9, 10)
d. R=[-3, 4]e. (-2, 0), (3, 0), (7, 0)
f. 4 times
Increasing and Decreasing Functions
Decreasing Constant Increasing
Example 2: Determine the open intervals on which each function is increasing, Example 2: Determine the open intervals on which each function is increasing, decreasing, or constant.decreasing, or constant.
a. b.
c.
(-1, 2)
(1, -2)
f(x) = x3 - 3x
(0, 1) (2, 1)
23
201
01
)(
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t
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tf
Increasing: (-∞,∞)
Increasing:(-∞, -1) U (1,∞)
Decreasing:(-1, 1)
Increasing: (-∞, 0)Constant: (0, 2)Decreasing: (2, ∞)
Relative Max and Min ValuesRelative Max and Min ValuesRelativeMax
Relative Min
Example 3: Use a graphing utility to approximate the relative Example 3: Use a graphing utility to approximate the relative maximum/minimum of the functionsmaximum/minimum of the functions
Sol: Min (0.67, -3.33)
a) f(x) = 3xf(x) = 3x2 2 - 4x - 2- 4x - 2 b) f(x) = -x3+x
Sol: Min (-0.58, -0.38)Max (0.58, 0.38)
Khan Academy Khan Academy
Recognizing Odd and Even FunctionsRecognizing Odd and Even Functions
Connection between even and odd Connection between even and odd numbers and functionsnumbers and functions
Even and Odd Functions: GraphicallyEven and Odd Functions: Graphically
Even FunctionSymmetric to y-axis
Odd FunctionSymmetric to origin
Not a Function:Symmetric to x-axis
Even and Odd Functions: Algebraically Even and Odd Functions: Algebraically
xxfa )(. xxxgb 3)(. 1)(. 2 xxhc
Even Function: A function f is even if, for each x in the domain of f, f(-x) = f(x).
Odd Function: A function f is odd if, for each x in the domain of f, f(-x) = -f(x)
Example 7: Determine whether each function is even, odd or neither.
1)(. 3 xxfd
xxxf )(
)(
)()()(
3
3
3
xx
xx
xxxg
1
1)()(2
2
x
xxh
)1(
1
1)()(
3
3
3
x
x
xxf
EVEN
ODDEVEN
NEITHER
HomeworkHomework
Pg96Pg96 #1-6all, #13-18 all, 21,22, 48, 50#1-6all, #13-18 all, 21,22, 48, 50