1.4 Analyzing Graphs of Functions

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)Written as [2, 5)Range [ -3, 8]

open and close becomes a big

deal(2, -3) (5,-1)

Interval Notation

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

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

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

f(x) = x3 -4x2 + 2x – 8

f(x) = x2(x - 4) + 2(x - 4)

Group factoring

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

We only worry about the numerator. 0 = 2a – 6 a = 3

“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

Here

f(2) f(3)

x1 x2 x3 x4

Constant Function

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

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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Homework

Homework Day 2

Page 47 – 50

#17, 23, 33, 37, 49, 55, 57, 61, 63, 83, 89