1. Domain1. Domain
2. Intercepts2. Intercepts
3. Asymptotes3. Asymptotes
4. Symmetry4. Symmetry
5. First Derivative5. First Derivative
6. Second Derivative6. Second Derivative
7. Graph 7. Graph
DomainDomain
Denominator can not be zeroDenominator can not be zero
D=(-oo,-3)U(-D=(-oo,-3)U(-3,3)U(3,oo)3,3)U(3,oo)
Nonnegatives under even rootsNonnegatives under even roots
1-x1-x22 >= 0 >= 0
x not 0x not 0
D = [-1, 0) U (0, 1]D = [-1, 0) U (0, 1]
2
4
9y
x
21 xy
x
The domain of y =The domain of y =is x -1 or D = is x -1 or D =
A.A. TrueTrue
B.B. FalseFalse
3
2 6
( 1)
x
x
, 1 1,
The domain of y =The domain of y =is x -1 or D = is x -1 or D =
A.A. TrueTrue
B.B. FalseFalse
3
2 6
( 1)
x
x
, 1 1,
DomainDomain
Denominator can not be zeroDenominator can not be zero
Nonnegatives under even rootsNonnegatives under even roots
1-x1-x22>=0 1-x>=0>=0 1-x>=0
and 1+x>=0 and 1+x>=0
D=[-1, 0) U (0, 1]D=[-1, 0) U (0, 1]
2
4
9y
x
21 xy
x
InterceptsIntercepts
Set x = 0 and solve for ySet x = 0 and solve for y
Set y = 0 and solve for xSet y = 0 and solve for x
SymmetrySymmetry
f(-x) = f(x) => Even functionf(-x) = f(x) => Even function
Symmetry about the y axisSymmetry about the y axis
f(-x) = -f(x) => Odd functionf(-x) = -f(x) => Odd function
Symmetry about the originSymmetry about the origin
AsymptotesAsymptotes
Denominator = 0 when x = cDenominator = 0 when x = c
x = c is an asymptotex = c is an asymptote
y = c is an asymptotey = c is an asymptote
lim ( )x
f x c
First derivativeFirst derivative
Find the critical pointsFind the critical points
Max, min, or neitherMax, min, or neither
Increasing or decreasingIncreasing or decreasing
Second derivative Second derivative
ConcavityConcavity
Inflection pointsInflection points
GraphGraph
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Series2
1. Domain1. Domain
2. Intercepts2. Intercepts
3. Asymptotes3. Asymptotes
4. Symmetry4. Symmetry
5. First Derivative5. First Derivative
6. Second Derivative6. Second Derivative
7. Graph 7. Graph
2
2
( 1)
1
xy
x
DomainDomain
Denominator can not be zeroDenominator can not be zero
No square roots so Domain =No square roots so Domain =
Domain = R Domain = R
2
2
( 1)
1
xy
x
The domain is all real The domain is all real numbers or (-oo, +oo)numbers or (-oo, +oo)
A.A. TrueTrue
B.B. FalseFalse2
2
( 1)
1
xy
x
The domain is all real The domain is all real numbers or (-oo, +oo)numbers or (-oo, +oo)
A.A. TrueTrue
B.B. FalseFalse2
2
( 1)
1
xy
x
InterceptsIntercepts
Set x = 0 and solve for ySet x = 0 and solve for y
y intercepty intercept
Set y = 0 and solve for xSet y = 0 and solve for x
x interceptx intercept
2
2
( 1)
1
xy
x
InterceptsIntercepts
Set x = 0 and solve for ySet x = 0 and solve for y
y intercepty intercept
Set y = 0 and solve for xSet y = 0 and solve for x
x interceptx intercept
2
2
( 1)
1
xy
x
InterceptsIntercepts
Set x = 0 and solve for ySet x = 0 and solve for y
y = 1y = 1
Set y = 0 and solve for xSet y = 0 and solve for x
(x + 1)(x + 1)22 = 0 when x = -1 = 0 when x = -1
2
2
( 1)
1
xy
x
SymmetrySymmetry
f(-x) not equal f(x) => Not even f(-x) not equal f(x) => Not even functionfunction
f(-x) = (-x+1)f(-x) = (-x+1)22/(1+x/(1+x22))
-f(x) = (-x-f(x) = (-x22-2x-1)/(1+x-2x-1)/(1+x22) not equal f(-x)) not equal f(-x)
Not an odd functionNot an odd function
No symmetry about the originNo symmetry about the origin
2
2
( 1)
1
xy
x
AsymptotesAsymptotes
Where is the denominator zero?Where is the denominator zero?
lim ( )x
f x c
2
2
( 1)
1
xy
x
2
2
2 1lim
1x
x x
x
The denominator is The denominator is zero when x = -1.zero when x = -1.
A.A. TrueTrue
B.B. FalseFalse2
2
( 1)
1
xy
x
The denominator is The denominator is zero when x = -1.zero when x = -1.
A.A. TrueTrue
B.B. FalseFalse2
2
( 1)
1
xy
x
y =y =
y’ =y’ =
= =
2
2
( 1)
1
x
x
2 2
2 2
(1 )2( 1) ( 1) 2
(1 )
x x x x
x
2
2 2 2 2
( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )
(1 ) (1 )
x x x x x x
x x
What is the absolute What is the absolute value of both critical value of both critical points?points?
2
2
( 1)
1
xy
x
2
2 2 2 2
( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )
(1 ) (1 )
x x x x x x
x x
What is the absolute What is the absolute value of both critical value of both critical points?points?
1.01.0
0.10.12
2
( 1)
1
xy
x
2
2 2 2 2
( 1)[(1 )2 ( 1)2 ] ( 1)(2 2 )
(1 ) (1 )
x x x x x x
x x
Increasing?Increasing?
y’ =y’ =
y’(-2) < 0 y’(0) >0 y’(2) < 0 y’(-2) < 0 y’(0) >0 y’(2) < 0
2 2
( 1)(2 2 )
(1 )
x x
x
Where is it increasing?Where is it increasing?
A.A. (1, +oo)(1, +oo)
B.B. (-oo, -1)(-oo, -1)
C.C. (-1, 1)(-1, 1)
2
2
( 1)
1
xy
x
Where is it increasing?Where is it increasing?
A.A. (1, +oo)(1, +oo)
B.B. (-oo, -1)(-oo, -1)
C.C. (-1, 1)(-1, 1)
2
2
( 1)
1
xy
x
First derivativeFirst derivativeFind the critical pointsFind the critical points
x = -1, 1x = -1, 1y = 0, 2y = 0, 2Decreasing on (-oo, -1) U (1, +oo)Decreasing on (-oo, -1) U (1, +oo)Increasing on (-1, 1)Increasing on (-1, 1)Local min at x=-1 and local max at Local min at x=-1 and local max at x=1x=1
2
2 2
2(1 )'( )
(1 )
xf x
x
2
2
( 1)
1
xy
x
ConcavityConcavity
Find the inflection pointsFind the inflection points
x = 0 , -root(3), root(3)x = 0 , -root(3), root(3)
2
2 3
4 ( 3)''( )
(1 )
x xf x
x
2
2
( 1)
1
xy
x
ConcavityConcavity
Inflection pts at x = 0 , Inflection pts at x = 0 ,
y = 1, [root(3) + 1]y = 1, [root(3) + 1]22/4 , [-root(3) + /4 , [-root(3) + 1]1]22/4/4
1.87 0.131.87 0.13
2
2
( 1)
1
xy
x
3