Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12 Cumulative Review
3x
x 2
6. f ( x)
= , g ( x) = 2x+
1
−
a. g ( ) ( )
f ( )
( )
2 = 2 2 + 1=
5
35 15
5 = = = 5
5−
2 3
( f g)( ) f ( g( )) f ( )
b. f ( )
2 = 2 = 5 = 5
( )
34 12
4 = = = 6
4−
2 2
g ( 6) = 2( 6)
+ 1=
13
( g
f )( ) g( f ( )) g( )
4 = 4 = 6 = 13
c. ( f g)( x) = f ( g( x)
)
32 ( x + 1)
=
( x + ) −
2 1 2
6x
+ 3
=
2x
−1
d. To determine the domain of the composition
( f g)( x)
, we start with the domain of g
and exclude any values in the domain of g
that make the composition undefined.
g x is defined for all real numbers and
( )
( f g)( x)
is defined for all real numbers
except
1
x = . Therefore, the domain of the
2
⎧ 1 ⎫
composite ( f g)( x)
is
e. ( g
f )( x)
⎛ 3x
⎞
= 2⎜
⎟+
1
⎝ x − 2 ⎠
6x
= + 1
x − 2
6x+ x−2
=
x − 2
7x
− 2
=
x − 2
⎨xx≠
⎬
⎩ 2⎭ .
f. To determine the domain of the composition
( g f )( x)
, we start with the domain of f
and exclude any values in the domain of f
that make the composition undefined.
f ( x ) is defined for all real numbers except
2 g f x is defined for all real
x = and ( )( )
numbers except x = 2 . Therefore, the
domain of the composite ( g f )( x)
is
{ x| x ≠ 2}
.
g. g ( x) = 2x+
1
1
2
y = 2x+
1
x = 2y
+ 1
x − 1=
2y
( x 1)
− = y
−
g ( x) = ( x−
1)
2
−1
The domain of g ( x)
1 1
numbers.
3x
f x =
x − 2
3x
y =
x − 2
3y
x =
y − 2
x( y− 2)
= 3y
xy− 2x = 3y
xy− 3y = 2x
y( x− 3)
= 2x
2x
y =
x − 3
−1 2x
f ( x)
=
x − 3
f
h. ( )
−1
The domain of ( x)
is the set of all real
is { x| x ≠ 3}
.
7. Center: (0, 0); Focus: (0, 3); Vertex: (0, 4);
Major axis is the y-axis; a = 4; c = 3 .
2 2 2
Find b: b = a − c = 16 − 9 = 7 ⇒ b=
7
Write the equation using rectangular coordinates:
2 2
x y
+ = 1
7 16
Parametric equations for the ellipse are:
x = 7cos π t ; y = 4sin π t ; 0≤t
≤ 2
( ) ( )
1285
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