R

1
R
R
-8, ?)
R
0, ?)
4, ?)
0, ?)
(- ?, 3
R \ ½
R \ 2
R \ 1
R \ -3, 0
R
(- ?, 4 U 2, ?)
(- 3, ?)
(- ?, -1) U 0, ?)
0, ?)
2
discontinuous infinite
discontinuous removable
continuous
discontinuous removable
discontinuous jump
continuous
discontinuous - jump
continuous
continuous
continuous
discontinuous - infinite
discontinuous - infinite
3
(3x4)(xlt1)(x-1)(xgt1)
jump
(x31)(x?0)
(2)(x0)
removable
(3x2)(xlt-2)(2x)(xgt-2)
(xlt1)(11-x2)(xgt1)
jump
4
decr (- ?, 0 incr 0, ?)
incr (- ?, ?)
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 ( 0, ? ) incr ( - ?, 0 )
decr ( - ?, ? )
decr (- ?, -8 incr 8, ? )
decr ( 2, ? ) incr ( - ?, 2) constant
-2, 2
decr ( - ?, 0 incr 0, 3 ) constant 3,
? )
decr ( - ?, 7 ) decr ( 7, ? )
5
bounded below b 0
bounded below b 1
unbounded
bounded above B 0
Left branch bounded above B 5
bounded b -1, B 1
unbounded
Right branch bounded below b 5
bounded below b 0
bounded below b -1
bounded below b 0
bounded above B 0
6
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)
7
Even
Odd
Even
Even
Neither
Odd
Even
Neither
Odd
Even
8
horizontally
vertically
will not cross
asymptotes
tan and cot
x 2
x -1
y 0
End behavior
Limit notation
9
Horizontal y 0 Vertical x 3
Horizontal y 0 Vertical x 2, -2
Vertical x - 3
10
Yes
Each x-value has only 1 y-value
( - ?, -1 ) U (-1, 1) U (1, ? )
( - ?, 0) U 3, ? )
Infinite discontinuity
Decreasing (- ?, -1), (-1, 0
Increasing ( 0, 1), (1, ? )
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
11
Yes
Each x-value has only 1 y-value
( - ?, ? )
0, ? )
continuous
Decreasing (- ?, 0
Increasing 0, ? )
Bounded below b 0
Absolute min 0 at x 0
Neither even or odd
none
( - ?, -3 U 7, ? )
12
10 Basic Functions
13
In-class Exercise Section 1.3

• Domain
• Range
• Continuity
• Increasing
• Decreasing
• Boundedness
• Extrema
• Symmetry
• Asymptotes
• End Behavior

14
f(x) g(x)
f(x) g(x)
f(x)g(x)
f(x)/g(x), provided g(x) ? 0
3x3 x2 6
3x3 x2 8
3x5 3x3 7x2 7
x2 (x 4) x2 x 4
15
x2
sin(x)
, , 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
16
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17
inverse
functions
horizontal line test
original relation
Graph is a function (passes vertical line
test. Inverse is also a function (passes
horizontal line test.)
Graph is a function (passes vertical line
test. Inverse is not a function (fails horizontal
line test.)
both vertical and horizontal
one-to-one function
line test like A
is paired with a unique y
is paired with a unique x
inverse function
f 1 (b) a, iff f(a) b
f 1
18
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19
D ( - ?, ? ) R ( - ?, ? )
D 0, ? ) R 0, ? )
D 0, ? )
D ( - ?, ? )
D ( - ?, - 2) U ( -2, ? ) R ( - ?, 1) U
(1, ? )
D ( - ?, 1) U (1, ? )
20
inside function
outside function
x2 1
x2
21
( - ?, ? )
( - ?, ? )
( - ?, ? )
( - ?, - 5) U ( - 5, ?)
( - ?, ? )
( - ?, ? )
f(x) and g(x) are inverses
22
Yes
passes horizontal line test
Yes
D ( - ?, 0 ) U ( 0, ? )
R ( - ?, 4 ) U ( 4, ? )
D ( - ?, 4 ) U ( 4, ? )
23
D ( - ?, - 2 ) U ( - 2, 1 ) U ( 1, ? )
D ( - ?, 2/3 ) U ( 2/3, 1 ) U ( 1, ? )
D (- ?, - 2) U (- 2, 1) U ( 1, ? )
D ( - ?, 0 ) U ( 0, ? )
24
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
left c units
f(x c)
25
reflections
negate the entire function y f(x)
negate x within the function y f(-x)
26
multiply c to the entire function
Stretch if c gt 1 Shrink if c lt 1
multiply c to x within the function
Stretch if c gt 1 Shrink if c lt 1
A reflection combined with a distortion
complete any stretches, shrinks or reflections
first
complete any shifts (translations)
27
y 1/x
4
y x, y x3, y 1/x, y ln (x)
y sqrt(x)
Answers
y ln(x)
y 2sin(0.5x)
Stretch by 8
Shrink ½
Stretch by 2
Shrink by 1/8