Slide 2.1-2 Chapter 1 Test Mean: 84 Median: 89 Breakdown: 22 A’s 12 B’s 4 C’s 4 D’s 3 F’s Copyright © 2009 Pearson Education, Inc.

Slide 2.1-2

Chapter 1 Test

Mean: 84

Median: 89

Breakdown:

22 A’s

12 B’s

4 C’s

4 D’s

3 F’s

Copyright © 2009 Pearson Education, Inc.


CHAPTER 2:More on Functions

2.1 Increasing, Decreasing, and Piecewise Functions; Applications2.2 The Algebra of Functions2.3 The Composition of Functions2.4 Symmetry and Transformations 2.5 Variation and Applications


2.1 Increasing, Decreasing, and Piecewise

Functions; Applications

Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.

Given an application, find a function that models the application; find the domain of the function and function values, and then graph the function.

Graph functions defined piecewise.

Slide 2.1-5Copyright © 2009 Pearson Education, Inc.

Increasing, Decreasing, and Constant Functions

On a given interval, if the graph of a function rises from left to right, it is said to be increasing on that interval. If the graph drops from left to right, it is said to be decreasing.

If the function values stay the same from left to right, the function is said to be constant.


Definitions

A function f is said to be increasing on an open interval I, if for all a and b in that interval,

a < b implies f(a) < f(b).


Definitions continued

A function f is said to be decreasing on an open interval I, if for all a and b in that interval,

a < b implies f(a) > f(b).


A function f is said to be

constant on an open interval I,

if for all a and b in that interval, f(a) = f(b).

Definitions continued


Relative Maximum and Minimum Values

Suppose that f is a function for which f(c) exists for some c in the domain of f. Then:

f(c) is a relative maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x c; and

f(c) is a relative minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x c.


Relative Maximum and Minimum Values

y

xc1 c2 c3

Relative minimum

Relative maximum f


Applications of Functions Many real-world situations can be modeled by functions.

ExampleA man plans to enclose a rectangular area using 80 yards of fencing. If the area is w yards wide, express the enclosed area as a function of w.

SolutionWe want area as a function of w. Since the area is rectangular, we have A = lw. We know that the perimeter, 2 lengths and 2 widths, is 80 yds, so we have 40 yds for one length and one width. If the width is w, then the length, l, can be given by l = 40 – w.

Now A(w) = (40 – w)w = 40w – w2.


Functions Defined Piecewise

For the function defined as:

find f (-3), f (1), and f (5).

2 , for 0,

( ) 4, for 0 2,

1, for 2,

x x

f x x

x x

Some functions are defined piecewise using different output formulas for different parts of the domain.

Since –3 0, use f (x) = x2: f (–3) = (–3)2 = 9.

Since 0 < 1 2, use f (x) = 4: f (1) = 4.

Since 5 > 2 use f (x) = x – 1: f (5) = 5 – 1 = 4.


c) We graph f(x) = only for inputs x greater than 2.

Functions Defined PiecewiseGraph the function defined as:

a) We graph f(x) = 3 only for inputs x less than or equal to 0.

b) We graph f(x) = 3 + x2 only for inputs x greater than 0 and less than or equal to 2.

2

3 for 0

( ) 3 for 0 2

1 for 22

x

f x x x

xx

f(x) = 3, for x 0

f(x) = 3 + x2, for 0 < x 2

( ) 1for 22

xf x x

12

x


= the greatest integer less than or equal to x.

Greatest Integer Function

5

5.2

15

8

The greatest integer function pairs the input with the greatest integer less than or equal to that input.

–5

0

0.2

1

7

0

3

3.2

13

8

3

Slide 2.1-15

Determine the intervals on which the function is (a) increasing, (b) decreasing, and © constant


Slide 2.1-16

Determine the domain and range of the function


Slide 2.1-17

Determine any relative maxima or minima


Slide 2.1-18

Determine the intervals on which the function is (a) increasing, (b) decreasing, and © constant


Slide 2.1-19

Determine the domain and range of the function


Slide 2.1-20

Determine any relative maxima or minima


Slide 2.1-21

Graph the function. Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima


53)( xxf

Slide 2.1-22

Graph the function using the given viewing window. Find the intervals on which the function is increasing or decreasing and find any relative maxima or minima.


]15,20,7,3[

496)( 23

xxxxf

Slide 2.1-23

For each piecewise function, find the specified function values.


)4(

)1(

)0(

)5(

1,2

15,1

5,183

)(

h

h

h

h

forxx

xfor

forxx

xh

Slide 2.1-24

http://www.education.com/study-help/article/pre-calculus-help-practice-functions/


Slide 2.1-2 Chapter 1 Test Mean: 84 Median: 89 Breakdown: 22 A’s 12 B’s 4 C’s 4 D’s 3 F’s Copyright © 2009 Pearson Education, Inc.

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