R
R
R
{ [ -8, ) }
{ [ 4, ) }
{ [ 0, ) }
{ (- , 3 ] }
{ [ 0, ) }
{R\{2}}
{R\{½}}
{R\{1}}
{ R \ { -3, 0 } }
R
{ (- , 4 ] U [ 2, ) }
{ (- 3, ) }
{ (- , -1) U [ 0, ) }
Pre-Calculus
{ [ 0, ) }
9/5/2006
continuous
continuous
discontinuous
infinite
discontinuous
removable
discontinuous - jump
discontinuous
removable
discontinuous
jump
continuous
continuous
continuous
Pre-Calculus
discontinuous - infinite
discontinuous - infinite
9/5/2006
(3x+4)(x<1)+(x-1)(x>1)
jump
(x^3+1)(x0)+
(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: [ - 1, 1 ]
incr: (- , -1 ], [ 1, )
decr: (- , 0 ]
incr: [ 0, )
decr: [ 3, 5 ], incr: [ , 3 ] decr: [ 3, ), incr: ( 0 ]
constant: [ 5, )
constant: [ 0, 3)
decr: ( - ,  )
decr: (- , -8 ]
incr: [ 8,  )
Pre-Calculus
decr: (- , 0 ]
incr: [ 0, )
decr: ( - , 0 ]
incr: [ 0, 3 )
constant: [ 3,  )
decr: ( 0,  )
incr: ( - , 0 )
decr: ( 2,  )
incr: ( - , 2)
constant: [ -2, 2 ]
decr: ( - , 7 )
decr: ( 7,  )
9/5/2006
unbounded
Left branch:
bounded above
B=5
Right branch:
bounded below
b=5
unbounded
bounded below
b=0
bounded above
B=0
bounded below
b=0
bounded below
b=0
Pre-Calculus
bounded below
b=1
bounded
b= -1, B = 1
bounded below
b = -1
bounded above
B=0
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,
Pre-Calculus
f(-x) = - f(x)
9/5/2006
Odd
Even
Even
Odd
Pre-Calculus
Even
Neither
Even
Neither
Even
Odd
9/5/2006
horizontally
vertically
will not cross
asymptotes
tan and cot
x=2
x = -1
y=0
End behavior
Limit notation
lim f(x)
x  
Pre-Calculus
lim f(x)
x  
lim f(x)  0
x  
lim f(x)  0
x  
9/5/2006
Horizontal: y = 0
Vertical: x = 2, -2
Horizontal: y = 0
Vertical: x = 3
x  
lim f(x)  3
lim f(x)  1
x  
lim f(x)  0
Vertical: x = - 3
lim f(x)  5
lim f(x)  5
lim f(x)  0
x  
lim f(x)  7
x  
Pre-Calculus
x  
x  
lim f(x)  0
x  
lim f(x)  
x  
x  
lim f(x)  1
x  
lim f(x)  4
x  
lim f(x)  4
x  
9/5/2006
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
lim f(x)  0
lim f(x)  0
x  
Pre-Calculus
x  
9/5/2006
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
lim f(x)  
x  
lim f(x)  
x  
{ ( - , -3 ] U [ 7,  ) }
Pre-Calculus
9/5/2006
10 Basic Functions
f(x)  x
3
1
f(x) 
x
f(x)  x
f(x)  cos x
f(x)  e
f(x)  x
x
f(x)  ln x
Pre-Calculus
f(x)  sin x
f(x)  x
f(x)  x 2
9/5/2006
In-class Exercise
Section 1.3
•Domain
•Range
•Continuity
•Increasing
•Decreasing
•Boundedness
•Extrema
•Symmetry
•Asymptotes
Pre-Calculus
•End Behavior
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 + 6
3x3 – x2 + 8
3x5 – 3x3 + 7x2 – 7
3x 3  7
x2  1
Pre-Calculus
x2 – (x + 4) = x2 – x – 4
9/5/2006
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
Pre-Calculus
4x – 9
9/5/2006
x2
4
4x
1  2x
1
1
x
1/ 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.)
Graph is a
function
(passes vertical
line test.
Inverse is not a
function (fails
horizontal line
test.)
both vertical and horizontal
line test like A
is paired with a unique y
one-to-one function
is paired with a unique x
f –1
Pre-Calculus
inverse function
f –1 (b) = a, iff f(a) = b
9/5/2006
Pre-Calculus
9/5/2006
D: { ( - ,  ) }
R: { ( - ,  ) }
D: { [ 0,  ) }
R: { [ 0,  ) }
x  2y  3
f 1(x) 
x y
x3
2
f 1(x)  x 2
D: { ( - ,  ) }
D: { [ 0,  ) }
D: { ( - , - 2) U ( -2,  ) }
R: { ( - , 1) U (1,  ) }
x
Pre-Calculus
y
y2
2x
f (x) 
x  1 D: { ( - , 1) U (1,  ) }
1
9/5/2006
inside function
outside function
f(g(x))  f(x 2  1)  x 2  1  h(x)
x
x2 + 1
x2
x 1
f(g(x))  f(x 2 )  x 2  1  h(x)
g(x)  2x  1
f(x)  x 3
g(x)  x 2  5
f(x)  x
g(x)  x  1
f(x)  x 2  3x  4
g(x)  x  2
Pre-Calculus
f(x)  4x 7  5
9/5/2006
1
5
( x  5)  (2x  10)  x  5
2
2
1
( x  5)(2x  10)  x 2  5x  50
2
{ ( - ,  ) }
1
3
( x  5)  (2x  10)   x  15
2
2
1
(2x  10)  5  x  5  5  x
2
1
( x  5)
2
(2x  10)
1
2( x  5)  10)  x  10  10  x
2
Pre-Calculus
f(x) and g(x) are inverses
{ ( - ,  ) }
{ ( - ,  ) }
{ ( - , - 5) U
( - 5, ) }
{ ( - ,  ) }
{ ( - ,  ) }
9/5/2006
Yes
passes horizontal line test
Yes
D: { ( - , 0 ) U ( 0,  ) }
R: { ( - , 4 ) U ( 4,  ) }
g(x)  x 2  2
f(x)  x
Pre-Calculus
( 3,6.75)
(6.75, 3)
( 2,2)
( 1,0.25)
(0,0)
(1, .25)
(2, 2)
(3, 6.75)
(2, 2)
(0.25, 1)
(0,0)
( .25,1)
( 2,2)
( 6.75,3)
1
4
y
1
y
x4
x
1
f (x) 
x4
1
D: { ( - , 4 ) U ( 4,  ) }
f(g(x))  f(x 2  2)  (x 2  2)  h(x)
9/5/2006
3
f(g(x))  x  2
3
1
x2
g(f(x)) 
3
x
2
x 1
x
f
x 1
  (x)  3
g
x2
x
Pre-Calculus
3
y2
3

1 x
D: { ( - , - 2 ) U ( - 2, 1 ) U ( 1,  ) }
3x  3

3x  2
D: { ( - , 2/3 ) U ( 2/3, 1 ) U ( 1,  ) }
x 2  2x

3x  3
D: { (- , - 2) U (- 2, 1) U ( 1,  ) }
3  2x
y
x
D: { ( - , 0 ) U ( 0,  ) }
9/5/2006
add or subtract a constant to the entire function
f(x) + c
f(x) – c
up c units
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
y  (x  3)2  4
Pre-Calculus
9/5/2006
reflections
negate the entire function y = – f(x)
negate x within the function y = f(-x)
3x  1
 3x  1 
 2

f(x)   x  2 
x2  2
3(  x)  1 3x  1
 2
2
f(  x)  (  x)  2 x  2
3x  1
x2  2
Pre-Calculus
3x  1
x2  2
3x  1
x2  2
9/5/2006
multiply c to the entire function
c g f(x)
Stretch if c > 1
Shrink if c < 1
multiply c to x within the function
x
f  
c
Stretch if c > 1
Shrink if c < 1
A reflection combined
with a distortion
complete any stretches, shrinks or reflections first
complete any shifts (translations)
Pre-Calculus
9/5/2006
y = 1/x
4
y = x, y = x3, y = 1/x, y = ln (x)
A
n
s
w
e
r
s
y = ln(x)
y = 2sin(0.5x)
Stretch by 8
Pre-Calculus
y = sqrt(x)
Shrink by 1/8
Shrink ½
Stretch by 2
9/5/2006