AIM: WHAT IS A PIECEWISE DEFINED FUNCTION? HW P. 25 #37, 39, 95, 97, P. 39#43, 48 1 EX) Sketch the...

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AIM: WHAT IS A PIECEWISE DEFINED FUNCTION? HW P. 25 #37, 39, 95, 97, P. 39#43, 48

EX) Sketch the Piecewise-Defined Function BY HAND

2 3, 1

4, 1

x xf x

x x

x 2x+3 y

1 2(1)+3 5

0 2(0)+3 2

-1 2(-1)+3 1

For each function, start at the domain and work your way up/down

Note: f(x) is not connected at x = 1

x -x+4 y

1 -1+3 3

2 -2+3 2

3 -3+3 1

A Piecewise-Defined Function is: A function defined in pieces for certain x-values

2


2 1, 0

1, 0

x xf x

x x

EX) Evaluate the piecewise-defined function at x = -1 and x = 0

21 1 1 2f

0 0 1 1f

Graph the piecewise function on the calculator:

Divide out the domain; the domain acts as a Boolean – as long as it is true the value is 1, and when it is false the value is 0. Therefore, where the calculator tries to graph the function beyond the domain, it attempts to graph the function by 0, which is undefined.

3


EX) Sketch the Piecewise-Defined Function

2

0

0 1

1 1

x for x

f x x for x

for x

4


1 2f x x a) Write the function below as a Piecewise-Defined Function

b) Sketch the graph by hand

c) Find the area of the region bounded by the function and the line y = 5

1 2f x x 1 2f x x

1f x x

1 2

3

f x x

f x x

1 3

2 1 3

2 2

1

x x

x

x

x

1x 1x

1, 1

3, 1

x xf x

x x

5


1, 1

3, 1

x xf x

x x


c) Find the area of the region bounded by the function and the line y = 5

5 :

1

5 1

4

When y

When f x x

x

x

5 :

3

5 3

2

When y

When f x x

x

x

Split up the region into right triangles and determine the lengths of the legs.

3

3 3

1

2

13 3 4.5

213 3 4.5

.24.5 4 5 9region

A

A

A

4-2

For this particular example, it was not important to perform the step above, however there will be examples in which the lengths of the legs are not as evident. The above process would then be necessary to correctly determine the lengths of the legs or the distance between 2 points.

A1 A2

6


a) Write the function below as a Piecewise-Defined Function


c) Find the area of the region bounded by the function and the line y = 6.8

2f x x

7


2, 2

2, 2

x xf x

x x

6.8 :

1

6.8 2

4.8

When y

When f x x

x

x

6.8 :

2

6.8 2

8.8

When y

When f x x

x

x

6.8

2 2

2 1 2 1

2 24.8 8.8 6.8 6.8

184.96

13.6

d x x y y

d

d

d

13.6

16.8 13.6 46.24

2regionA

4.8-8.8

8




c) Find the area of the region bounded by the function and the x-axis from x=-4 to x=0

3f x x

A1 A2

0 :

3

0 0 3

0 3

When x

Use f x x

f

f

4 :

3

4 4 3

4 1

When x

Use f x x

f

f

3, 3

3, 3

x xf x

x x

1

2

13 3 4.5

211 1 0.5

2

A

A

1 2

4.5 0.5 5.0reg

regi

on

on

i

A A A

A

9




c) Find the area of the region bounded by the function and the x-axis from x = -3 to x = 5.

2 3 1f x x

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Review for next test. Know how to:

Graph piecewise functions by hand

Graph piecewise functions on calc

Re-write an absolute value equation as a piecewise function

Find the area under a specified region on the x and y axes

Write a function in terms of a specified variable (see function apps sheet)

2

2

4

4

xf x

x

2

1, 0

, 0 1

1, 1 2

3 , 2

x x

x xh x

x

x x

2 3, 0

3 , 0

x xh x

x x

Re-write as a piecewise function:

Graph by hand and on calc:

Find area between function and x-axis, from x = -5 to x = 5:

2

2

5, 1

4 3, 1

x xh x

x x x

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#39: Evaluate f(-2), f(1), and f(2) and graph 2

2

2, 1

2 2, 1

x xf x

x x

12


#95, 97: Write a piecewise-defined functions for each of the graphs:

#43: Sketch the graph of the piecewise-defined functions:

2 3, 0

3 , 0

x xf x

x x

#48: Sketch the graph of the piecewise-defined functions:

5, 3

2, 3 1

5 4, 1

x x

g x x

x x

AIM: WHAT IS A PIECEWISE DEFINED FUNCTION? HW P. 25 #37, 39, 95, 97, P. 39#43, 48 1 EX) Sketch the...

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