2.7 Piecewise Functions

p. 114

Up to now, we’ve been looking at functions represented by a single equation.

In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.

These are called piecewise functions.

1 ,13

1 ,12

xifx

xifxxf

•One equation gives the value of f(x) when x ≤ 1•And the other when x>1

Evaluate f(x) when x=0, x=2, x=4

2 ,12

2 ,2)(

xifx

xifxxf

•First you have to figure out which equation to use•You NEVER use both

X=0This one fitsInto the top equation

So:0+2=2f(0)=2

X=2This one fits hereSo:2(2) + 1 = 5f(2) = 5

X=4

This one fits hereSo:2(4) + 1 = 9f(4) = 9

Graph:

1 ,3

1 ,)( 2

321

xifx

xifxxf

•For all x’s < 1, use the top graph (to the left of 1)

•For all x’s ≥ 1, use the bottom graph (to the •right of 1)

1 ,3

1 ,)( 2

321

xifx

xifxxf

x=1 is the breakingpoint of the graph.

To the left is the topequation.

To the right is thebottom equation.

Graph:

2 ,1

2 ,32

32

)(xifx

xifxxf

Step Functions

43 ,432 ,321 ,210 ,1

)(

xifxifxifxif

xf

43,432,321,210,1

)(

xifxifxifxif

xf

Graph :

01,412,323,234,1

)(

xifxifxifxif

xf

2.7 (continued)Writing Equations for Piecewise Functions and Word Problemsp. 115

Writing a piecewise function given the graph.

The break in the graphIs at x=0.

To the left of x=0 the graph is part of the line thru (-2,0) & (0,2)

An equation of this line is:y = x + 2 y = x + 2

To the right of x=0, theGraph is part of the Line thru (0,0) & (2,2)

y = x +0y = x

Notice where the circle & dot are!!

The piecewise function is:

0 ,

0 ,2)(

xifx

xifxxf

(open dot)

(closed dot)

Write the equation of:

1 ,21

21

1 ,2)(

xifx

xifxxf

You have a summer job that pays time and a half for overtime. (If you work more than 40 hours) After that it is 1.5 times your hourly rate of $7.00/hr.

1. Write and graph a piecewise function that gives your weekly pay P in terms of the number hours you work h.

2. How much will you make if you work 45 hours?

From 0 to 40 hours you get $7.00/hr: 7h

over 40 you get: 7(40) + (1.5)(7)(h-40) = 7(40) + 10.5(h-40)=10.5h - 140The function is:

40 ,1405.10

400 ,7)(

hifh

hifhhP

So the graph is:

2. P(h) = 10.5h - 140

P(45) = 10.5(45) - 140

= $332.50 for 45 hours of work

Assignment

Assignment