hat04 0205
DESCRIPTION
TRANSCRIPT
Copyright © 2007 Pearson Education, Inc. Slide 2-1
2.5 Piecewise-Defined Functions
• The absolute value function is a simple example of a function defined by different rules over different subsets of its domain. Such a function is called a piecewise-defined function. – Domain of is with one rule on
and the other rule on
• Example Find each function value given the piecewise-defined function
Solution
(a)
(b)
(c)
xxf )( ),,( xxf )( ),0,(xxf )( ).,0[
.0 if2
1
0 if2)(
2 xx
xxxf
. ofgraph Sketch the (d) )3( (c) )0( (b) )3( (a) ffff
.123)3( thus,2)( rule theuse ,03 Since fxxf
.220)0( thus,2)( rule theuse ,00 Since fxxf
.5.4)9()3()3( thus,)( rule theuse ,03 Since21
21
21 22 fxxf
. ofgraph on the are )5.4,3( and (0,2), ),1,3( points The f:Note
Copyright © 2007 Pearson Education, Inc. Slide 2-2
2.5 The Graph of a Piecewise-Defined Function
(d) The graph of
Graph the ray choosing x so that with a solid endpoint (filled in circle) at (0,2). The ray has slope 1 and
y-intercept 2. Then, graph for This graph will be
half of a parabola with an open endpoint (open circle) at (0,0).
time.aat piece onedrawn is 0 if
2
1
0 if2)(
2
xx
xxxf
,2xy ,0x
221 xy .0x
Figure 51 pg 2-117
Copyright © 2007 Pearson Education, Inc. Slide 2-3
2.5 Graphing a Piecewise-Defined Function with a Graphing Calculator
• Use the test feature– Returns 1 if true, 0 if false when plotting the value of x
• In general, it is best to graph piecewise-defined functions in dot mode, especially when the graph exhibits discontinuities. Otherwise, the calculator may attempt to connect portions of the graph that are actually separate from one another.
Copyright © 2007 Pearson Education, Inc. Slide 2-4
2.5 Graphing a Piecewise-Defined Function
Sketch the graph of
Solution
For graph the part of the line to the left of, and including, the point (2,3). For graph the part of the line to the right of the point (2,3).
.2 if722 if1 )(
xxxxxf
1xy
72 xy
,2x,2x
Copyright © 2007 Pearson Education, Inc. Slide 2-5
2.5 The Greatest Integer (Step) Function
Example Evaluate for (a) –5, (b) 2.46, and (c) –6.5
Solution (a)
(b)
(c)
Using the Graphing Calculator
The command “int” is used by many graphing calculators for the greatest integer function.
integeran not is if than lessinteger greatest the
integeran is if xx
xx
52
7
( )f x x
x
Copyright © 2007 Pearson Education, Inc. Slide 2-6
2.5 The Graph of the Greatest Integer Function
– Domain:
– Range:
• If using a graphing calculator, put the calculator in dot mode.
||)( xxf ),(
},3,2,1,0,1,2,3,{}integeran is { xx
Figure 58 pg 2-124
Copyright © 2007 Pearson Education, Inc. Slide 2-7
2.5 Graphing a Step Function
• Graph the function defined by Give the domain and range.
Solution
Try some values of x.
. 121
xy
x -3 -2 -1 0 .5 1 2 3 4
y -1 0 0 1 1 1 2 2 3
}.,2,1,0,1,{ is range theand ),( isdomain The on. so
and ,2 ,42For .1 then ,20 if that Notice
yxyx
Copyright © 2007 Pearson Education, Inc. Slide 2-8
2.5 Application of a Piecewise-Defined Function
Downtown Parking charges a $5 base fee for parking through 1 hour, and $1 for each additional hour or fraction thereof. The maximum fee for 24 hours is $15. Sketch a graph of the function that describes this pricing scheme.
SolutionSample of ordered pairs (hours,price): (.25,5), (.75,5), (1,5), (1.5,6), (1.75,6).
During the 1st hour: price = $5During the 2nd hour: price = $6During the 3rd hour: price = $7
During the 11th hour: price = $15
It remains at $15 for the rest of the 24-hour period.
Plot the graph on the interval (0,24]. Figure 62 pg 2-127
Copyright © 2007 Pearson Education, Inc. Slide 2-9
2.5 Using a Piecewise-Defined Function to Analyze Data
Due to acid rain, The percentage of lakes in Scandinavia that lost their population of brown trout increased dramatically between 1940 and 1975. Based on a sample of 2850 lakes, this percentage can be approximated bythe piecewise-defined function f .
(a) Graph f .
(b) Determine the percentage of lakes that had lost brown trout by 1972.
19759601 if18)1960(15
32
19609401 if7)1940(20
11
)(
xx
xxxf
Copyright © 2007 Pearson Education, Inc. Slide 2-10
2.5 Using a Piecewise-Defined Function to Analyze Data
Solution(a) Analytic Solution: Plot the two endpoints and draw the line
segment of each rule.
Note: Even though there is an open circle at the point (1960,18)
from the first rule, the second rule closes it. Therefore, the
point (1960,18) is closed.
(1960,18)point theus gives ,18,1960
(1940,7)point theus gives ,7,1940
yx
yx:7)1940(20
11 xy
(1975,50)point theus gives ,50,1975
(1960,18)point theus gives ,18,1960
yx
yx:18)1960(15
32 xy
Figure 63 pg 2-128
Copyright © 2007 Pearson Education, Inc. Slide 2-11
2.5 Using a Piecewise-Defined Function to Analyze Data
Graphing Calculator Solution
(b) Use the second rule with x = 1972.
(percent) 6.4318)19601972(15
32)1972( f
By 1972, about 44% of the lakes had lost their population of brown trout.