Pre-Calculus

9/5/2006

R

R

{ [ 4, ) }

{ (- , 3 ] }

{ R \ { 2 } }

{ R \ { 1 } }

{ R \ { -3, 0 } }

R

{ (- 3, ) }{ (- , 4 ] U [ 2, ) }

{ (- , -1) U [ 0, ) } { [ 0, ) }

R

{ [ -8, ) }

{ [ 0, ) }

{ [ 0, ) }

{ R \ { ½ } }

Pre-Calculus

9/5/2006

continuous discontinuousinfinite

discontinuousremovable

continuous discontinuousremovable

discontinuousjump

discontinuous - jump

continuous

discontinuous - infinite

continuous

continuous

discontinuous - infinite

Pre-Calculus

9/5/2006

(3x+4)(x<1)+(x-1)(x>1)

jump

(x^3+1)(x0)+

(2)(x=0)

removable

(3+x2)(x<-2)+(2x)(x>-2)

(x<1)+(11-x2)(x>1)

jump

Pre-Calculus

9/5/2006

incr: (- , ) decr: (- , 0 ]incr: [ 0, )

decr: (- , 0 ]incr: [ 0, )

decr: [ - 1, 1 ]incr: (- , -1 ], [ 1, )

decr: [ 3, 5 ], incr: [ , 3 ]constant: [ 5, )

decr: [ 3, ), incr: ( 0 ]constant: [ 0, 3)

decr: ( - , )

decr: (- , -8 ]incr: [ 8, )

decr: ( - , 0 ]incr: [ 0, 3 )

constant: [ 3, )

decr: ( 0, )incr: ( - , 0 )

decr: ( 2, )incr: ( - , 2)

constant: [ -2, 2 ]

decr: ( - , 7 )decr: ( 7, )

Pre-Calculus

9/5/2006

unbounded bounded belowb = 0

bounded belowb = 1

unbounded bounded aboveB = 0

boundedb= -1, B = 1

bounded belowb = 0 bounded below

b = -1bounded below

b = 0bounded above

B = 0

Right branch:bounded below

b = 5

Left branch: bounded above

B = 5

Pre-Calculus

9/5/2006

y-axis

EVEN functions

The graph looks the same to the

left of the y-axis as it does to the right

For all x in the domain of f,

f(-x) = f(x)

x-axis

The graph looks the same above

the x-axis as it does below it

(x, - y) is on the graph whenever

(x, y) is on the graph

origin

ODD functions

The graph looks the same upside

Down as it does right side up

For all x in the domain of f,

f(-x) = - f(x)

Pre-Calculus

9/5/2006

Odd Even Even

Odd Even Neither

Even Neither

Even Odd

Pre-Calculus

9/5/2006

horizontally

vertically

will not cross

asymptotes

tan and cot

x = -1 x = 2

y = 0

End behavior

x

lim f(x)

x

lim f(x)

Limit notation

x

lim f(x) 0

x

lim f(x) 0

Pre-Calculus

9/5/2006

Vertical: x = - 3 Horizontal: y = 0Vertical: x = 2, -2

Horizontal: y = 0Vertical: x = 3

x

lim f(x) 5

x

lim f(x) 5

x

lim f(x) 3

x

lim f(x) 0

x

lim f(x) 1

x

lim f(x) 1

x

lim f(x) 0

x

lim f(x) 7

x

lim f(x) 0

x

lim f(x)

x

lim f(x) 4

x

lim f(x) 4

Pre-Calculus

9/5/2006

Yes

{ ( - , -1 ) U (-1, 1) U (1, ) }

Infinite discontinuity

Decreasing: (- , -1), (-1, 0 ]

Unbounded

Left piece: B = 0, Middle piece b = 3, Right piece B = 0

Local min at (0, 3)

Even

Horizontal: y = 0, Vertical: x = -1, 1

Each x-value has only 1 y-value

{ ( - , 0) U [ 3, ) }

Increasing: ([ 0, 1), (1, )

x

lim f(x) 0

x

lim f(x) 0

Pre-Calculus

9/5/2006

Yes

{ ( - , ) }

continuous

Decreasing: (- , 0 ]

Bounded below b = 0

Absolute min = 0 at x= 0

Neither even or odd

none

Each x-value has only 1 y-value

{ [ 0, ) }

Increasing: [ 0, )

x

lim f(x)

x

lim f(x)

{ ( - , -3 ] U [ 7, ) }

Pre-Calculus

9/5/2006

10 Basic Functions10 Basic Functions

3f(x) x f(x) sinx

f(x) cosx

f(x) x

f(x) x

2f(x) x

1

f(x)x

f(x) x

xf(x) e

f(x) lnx

Pre-Calculus

9/5/2006

In-class ExerciseSection 1.3

In-class ExerciseSection 1.3

•Domain

•Range

•Continuity

•Increasing

•Decreasing

•Boundedness

•Extrema

•Symmetry

•Asymptotes

•End Behavior

Pre-Calculus

9/5/2006

f(x) + g(x)f(x) – g(x)

f(x)g(x)f(x)/g(x), provided g(x) 0

3x3 + x2 + 63x3 – x2 + 83x5 – 3x3 + 7x2 – 7

x2 – (x + 4) = x2 – x – 4

3

2

3x 7

x 1

Pre-Calculus

9/5/2006

sin(x) x2

+, –, x,

applying them in order

the squaring function the sin function

function composition

f ○ g

(f ◦ g)(x) = f(g(x))

4x2 – 12x + 9

1

2x2 – 3

5

x4

4x – 9

Pre-Calculus

9/5/2006

x 2

4

4x

1 2x

1

1

x1/ x

2(x 2)

x 4

Pre-Calculus

9/5/2006

inverse

functions

horizontal line test

original relation

Graph is a function

(passes vertical line test.

Inverse is also a function (passes

horizontal line test.)

both vertical and horizontal

line test like A one-to-one function

is paired with a unique y

inverse function

is paired with a unique x

f –1 f –1 (b) = a, iff f(a) = b

Graph is a function

(passes vertical line test.

Inverse is not a function (fails horizontal line

test.)

Pre-Calculus

9/5/2006

Pre-Calculus

9/5/2006

D: { ( - , ) }R: { ( - , ) }

D: { [ 0, ) }R: { [ 0, ) }

D: { ( - , - 2) U ( -2, ) }R: { ( - , 1) U (1, ) }

D: { ( - , ) }D: { [ 0, ) }

D: { ( - , 1) U (1, ) }

x 2y 3

1 x 3

f (x)2

x y

1 2f (x) x

y

xy 2

1 2xf (x)

x 1

Pre-Calculus

9/5/2006

3

g(x) 2x 1

f(x) x

inside function

outside function

x2 + 1 x 2 2f(g(x)) f(x 1) x 1 h(x)x2

x 1 2 2f(g(x)) f(x ) x 1 h(x)

2g(x) x 5

f(x) x

2

g(x) x 1

f(x) x 3x 4

7g(x) x 2

f(x) 4x 5

Pre-Calculus

9/5/2006

{ ( - , ) } 1 5( x 5) (2x 10) x 52 2

f(x) and g(x) are inverses

1 3( x 5) (2x 10) x 152 2

21( x 5)(2x 10) x 5x 502

1( x 5)2(2x 10)

1(2x 10) 5 x 5 5 x2

1

2( x 5) 10) x 10 10 x2

{ ( - , ) }

{ ( - , ) }

{ ( - , - 5) U ( - 5, ) }

{ ( - , ) }

{ ( - , ) }

Pre-Calculus

9/5/2006

( 3,6.75)

( 2,2)

( 1,0.25)

(0,0)

(1, .25)

(2, 2)

(3, 6.75)

1x 4

y

1y

x 4

Yes

passes horizontal line test

Yes

(6.75, 3)

(2, 2)

(0.25, 1)

(0,0)

( .25,1)

( 2,2)

( 6.75,3)

D: { ( - , 0 ) U ( 0, ) }

R: { ( - , 4 ) U ( 4, ) } D: { ( - , 4 ) U ( 4, ) }

1 1f (x)

x 4

2g(x) x 2

f(x) x 2 2f(g(x)) f(x 2) (x 2) h(x)

Pre-Calculus

9/5/2006

D: { ( - , - 2 ) U ( - 2, 1 ) U ( 1, ) }

3x 2f(g(x))3

1x 2

3g(f(x))

x2

x 1

xf x 1(x)

3gx 2

D: { (- , - 2) U (- 2, 1) U ( 1, ) }

3

xy 2 D: { ( - , 0 ) U ( 0, ) }

3

1 x

3x 3

3x 2D: { ( - , 2/3 ) U ( 2/3, 1 ) U ( 1, ) }

2x 2x

3x 3

3 2x

yx

Pre-Calculus

9/5/2006

add or subtract a constant to the entire function

f(x) + c up c units

f(x) – c down c units

add or subtract a constant to x within the function

f(x – c) right c units

f(x + c) left c units

y cos(x) 5 y x 2

2y (x 3) 4

Pre-Calculus

9/5/2006

reflections

negate the entire function y = – f(x)

negate x within the function y = f(-x)

f(x) 2

3x 1

x 2

23x 1

x 2

f( x)

23( x) 1

( x) 2

23x 1

x 2

2

3x 1

x 2

2

3x 1

x 2 2

3x 1

x 2

Pre-Calculus

9/5/2006

multiply c to the entire function

Stretch if c > 1

Shrink if c < 1

multiply c to x within the function

A reflection combined with a distortion

complete any stretches, shrinks or reflections first

complete any shifts (translations)

x

fc

Stretch if c > 1

Shrink if c < 1

gc f(x)

Pre-Calculus

9/5/2006

Answers

Answers

y = 1/x4

y = x, y = x3, y = 1/x, y = ln (x)

y = sqrt(x)y = ln(x)

y = 2sin(0.5x)

Stretch by 8 Shrink ½

Shrink by 1/8 Stretch by 2