ACTIVITY 33

Review (Sections 3.1+3.2+3.3+3.4+3.5)

Problem 15:

Let .1

1)(

x

xxg

Evaluate

)2(g

)2(g

2

1g

ag

1ag

1g

21

21

3

1

21

21

21

21

1

3

21

1

21

1

21

22

21

22

212

212

2321

3

2*

2

1

3

1

a

a

1

1

11

11

a

a

11

11

a

a

a

a

2

11

11

!Undefined

Problem 23:

Let

-1x if

-1x if 2)(

2

x

xxxf Evaluate

)4(f

2

3f

1f

0f

1f

424 2 816 8

2

32

2

32

2

6

4

9

4

12

4

9

4

129

4

3

121 2 21 1

0

1

Problem 33:

2453)( xxxf find:

)(af

)( haf

2453 aa

2453 haha 22 2453 hahaha

22 484553 hahaha

Let

0. h where,)()(

h

afhaf

h

aahahaha )453()484553( 222

h

aahahaha 222 453484553

h

hahh 2485

h

hah 485 ha 485

Problems 39, 41, 47, and 53:

Find the domain of each function.

3

1)(

x

xf Since the numerator and denominator are polynomial, who’s domains are all real numbers, we need only be concerned with x – 3 = 0.

That is x = 3 is the only real number not in the domain. ,33,

1

2)(

2

x

xxf

Since the numerator and denominator are polynomial, who’s domains are all real numbers, we need only be concerned with x2 – 1 = 0.012 x

12 x

1x 1

,11,11,

52)( xxh We must have a nonnegative real number under the square root. Consequently, the domain is all x’s such that052 x

52 x

2

5x

2

5

,2

5

82)( 2 xxxgWe must have a nonnegative real number under the square root. Consequently, the domain is all x’s such that

0822 xx

2 ,42,

024 xxConsequently, the critical numbers are x = 4 and x = -2

4

Problems 1 and 19:

Sketch the graph of each function by first making a table of values.

x y

0 2

1 2

-1 22 2

-2 2

2)( xf

x y

0 2

1 0

-1 42 2

3 4

22)( xxf

Problems 41

0 if 1

0 if )(

xx

xxxf

Sketch the graph of the piecewise defined function

Problem 21:

Determine the average rate of change of f(x) = x3 − 4x2

between x = 0 and x = 10.

ab

afbf

)()(

arc

0a10b

010

)0()10(

ff

010

040104)10( 2323

010

04001000

10

600 60

Problem 23:

Determine the average rate of change of f(x) = 3x2 between x = 2 and x = 2 + h

ab

afbf

)()(

arc

2ahb 2

22

)2()2(

h

fhf h

h 22 23)2(3

h

hhh 122243 2

h

hh 12443 2

h

hh 1231212 2

h

hh 2312

h

hh

43 h 43

Problems 5 and 7:

Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f.

)(2 xfy The graph is stretched vertically and flipped over the x – axis.

)(2

1xfy The graph shrinks vertically and flipped over the

x – axis.

4

3)4( xfy

The graph is moved to the right 4 units and up ¾ units.

4

3)4( xfy The graph is moved to the left 4 units and down

¾ units.

)(2 xf )(2

1xf )2( xf

xf

2

1 xf

Problem 33:

Sketch the graph of f(x) = (x − 2)2, not by plotting points, but by starting with the graph of a standard function and applying transformations.

Problem 47:

Sketch the graph of f(x) = |x + 2| + 2, not by plotting points, but by starting with the graph of a standard function and applying transformations.

2,2

Problems 65

Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch the graph.

2)( xxf

2

1)(x

xf 2

1)(

xxf

2

1

x )(xf

Even functions are symmetric about the y – axis.

Even

x y

0 undefined

1 1

2 1/41/2 4

1/3 9

2

1)(x

xf

Problem 13:

Sketch the graph of y = 2x2 +4x +3 and state the coordinates of its vertex and its intercepts.

342 2 xxy

322 2 xxy

31122 2 xxy

32122 2 xxy

112 2 xy

khxay 2

1h1k

)1,1(VThe vertex lies in the second quadrant and the parabola is going up so there can be no x – intercepts!

intercepty

0x

30402 2 y 3 3,0

342 2 xxy

)1,1(V

3,0

112 2 xy

This 2 stretches the graph up

Problem 31:

Find the maximum or minimum value of

2

2

77

2

7491002

7

f

maximuma has functionour 7- a thesince f

a

bx

2at occure willmaximum The

72

49

14

49

2

7

is maximum thely,Consequent

4

343

2

343100

4

343

4

686

4

400

4

743

2749100 tttf