copyright © 2011 pearson education, inc. slide 2.5-1 2.5 piecewise-defined functions the absolute...

Copyright © 2011 Pearson Education, Inc. Slide 2.5-1

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


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


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.


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


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


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


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


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): (0.25,5), (0.75,5), (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.

Graph on the interval (0,24].Figure 62 pg 2-127


2.5 Using a Piecewise-Defined Function to Analyze Data

Due to acid rain, the percentage of lakes in Scandinavia that had 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 .

Determine the percent of lakes that had lost brown trout (a) by 1950 and (b) by 1972. Interpret your results.

19759601 if18)1960(15

32

19609401 if7)1940(20

11

)(

xx

xxxf


2.5 Using a Piecewise-Defined Function to Analyze Data

(a) Use the first rule with x = 1950.

(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.

11(1950) (1950 1940) 7 12.5

20f

By 1950, about 12.5% of the lakes had lost their population of brown trout.

copyright © 2011 pearson education, inc. slide 2.5-1 2.5 piecewise-defined functions the absolute...

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