Testing for even and odd functions. When the end points are included [ ]. When the end points are...
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Transcript of Testing for even and odd functions. When the end points are included [ ]. When the end points are...
![Page 1: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/1.jpg)
1.4 Analyzing Graphs of Functions
Testing for even and odd functions
![Page 2: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/2.jpg)
When the end points are included [ ]. When the end points are not included ( ).
(4,8) Domain from (2, -3) to (5, -1)Written as [2, 5)Range [ -3, 8]
open and close becomes a big
deal(2, -3) (5,-1)
Interval Notation
![Page 3: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/3.jpg)
Graphically using the Vertical line test.“ A set of points in a coordinate plane is
the graph of y as a function of x iff no vertical line intersect the graph at more than one point.”
Not a Function
Function
Testing for a function
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Zeros are the x’s that make f(x) = 0Find the zero of the function
f(x) = x3 -4x2 + 2x - 8
How do you find them?
Zeros of a Function
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Zeros are the x’s that make f(x) = 0Find the zero of the function
f(x) = x3 - 4x2 + 2x - 8
How do you find them?
Factoring would work
Zeros of a Function
![Page 6: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/6.jpg)
f(x) = x3 -4x2 + 2x – 8
f(x) = x2(x - 4) + 2(x - 4)
Group factoring
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f(x) = x3 -4x2 + 2x – 8
f(x) = x2(x - 4) + 2(x - 4)
f(x) = (x – 4)(x2 + 2)
0 = (x – 4) and 0 = (x2 + 2), 4 = x - 2 = x2
thus the only real answer is x = 4
Group factoring
![Page 8: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/8.jpg)
We only worry about the numerator. 0 = 2a – 6 a = 3
![Page 9: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/9.jpg)
“Increasing” function x1<x2 implies f (x2)>f (x1)
“Decreasing” functionx3<x4 implies f (x3)>f (x4)
f(2) f(3)
x1 x2 x3 x4
Increasing and Decreasing Function
![Page 10: Testing for even and odd functions. When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1)](https://reader035.fdocuments.us/reader035/viewer/2022072014/56649eaa5503460f94baf8f2/html5/thumbnails/10.jpg)
Here
f(2) f(3)
x1 x2 x3 x4
Constant Function
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Over a Given Interval
Minimum is the lowest point Maximum is the highest point.
This will lead to the “Extreme Value Theorem”
Definition of Relative Minimum and Maximum
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EVEN function is where f(x) = f(- x)Odd function is where f(- x) = - f(x)
Let g(x) = x3 + x thus ( -x)3 + (- x) so - x3 – x ; - g(x) = - (x3 + x)
It is then Odd
f(x) = x4 + 2 thus f(-x) = (-x)4 + 2 ; x4 + 2
which is the same as f(x) It is then Even
Even and Odd Functions
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Page 47-50# 2, 10, 16, 22, 32, 54, 60, 62, 66, 86
Homework
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Homework Day 2
Page 47 – 50
#17, 23, 33, 37, 49, 55, 57, 61, 63, 83, 89