PreCalculus
Chapter 2 Review
1.
Name_
Find the domain of :
A.
B.
f ( x) x 1
2.
Given
3.
Given
4.
Determine algebraically whether
5.
Determine algebraically whether
8
f ( x)
x2
, find the difference quotient
f (x) 2x 2 x 3
C.
3
f ( x) 2
x 3 x 10
f ( x h)
238
CHAPTER 4 Polynomial and Rational Functions
4.6 Assess Your Understanding
Are You Prepared?
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
1. Find the sum and the product of the complex number
228
CHAPTER 4 Polynomial and Rational Functions
Using the Theorem for Finding Bounds on Zeros
EXAMPLE 7
Find a bound on the real zeros of each polynomial function.
(a) f1x2 = x5 + 3x3  9x2 + 5
Solution
(b) g1x2 = 4x5  2x3 + 2x2 + 1
(a) The leading coeff
SECTION 4.6 Complex Zeros; Fundamental Theorem of Algebra
Now, because p is a factor of the first n terms of this
equation, p must also be a factor of the term a0qn. Since p is
not a factor of q (why?), p must be a factor of a0. Similarly, q
must be a fac
218
CHAPTER 4 Polynomial and Rational Functions
Concepts and Vocabulary
x
is above the
x  3
xaxis for x 6 0 or x 7 3, so the solution set of the inequality
x
0 is 5xx 0 or x 36.
x  3
4. True or False The graph of f1x2 =
3. True or False A test number
SECTION 4.5 The Real Zeros of a Polynomial Function
223
Replacing x by c, we find that
f1c2 = 1c  c2q1c2 = 0 # q1c2 = 0
This completes the proof.
EXAMPLE 2
Using the Factor Theorem
Use the Factor Theorem to determine whether the function
f1x2 = 2x3  x2
208
CHAPTER 4 Polynomial and Rational Functions
STEP 4: Since x + 2 is the only factor of the denominator of R1x2 in lowest terms, the
graph has one vertical asymptote, x =  2. However, the rational function is
undefined at both x = 2 and x =  2. Graph
SECTION 4.3 The Graph of a Rational Function
54. Minimizing Surface Area United Parcel Service has
contracted you to design an open box with a square base
that has a volume of 5000 cubic inches. See the illustration.
213
56. Material Needed to Make a Drum
SECTION 4.2 Properties of Rational Functions
193
Exploration
Graph each of the following rational functions:
R(x) =
1
x  1
R(x) =
1
(x  1)2
R(x) =
1
(x  1)3
R(x) =
1
(x  1)4
Each has the vertical asymptote x = 1. What happens to the value of R(x) as x
178
CHAPTER 4 Polynomial and Rational Functions
Solution
The yintercept of f is f102 =  6. We can eliminate the graph in Figure 19(a), whose
yintercept is positive.
We dont have any methods for finding the xintercepts of f, so we move on to
investigat
203
SECTION 4.3 The Graph of a Rational Function
Table 12
1
0
Interval
( q ,  1)
( 1, 0)
Number chosen
2

Value of R
R( 2) = 
3
2
Location of graph
Point on graph
1
x
(0, 1)
(1, q )
1
2
2
1
3
R a b =
2
2
3
1
Ra b = 2
2
R(2) =
Below xaxis
Above x
188
CHAPTER 4 Polynomial and Rational Functions
5. Obtain a graph of the function for the values of a, b, and c in the following table. Conjecture a relation between the degree of a
polynomial and the number of turning points after completing the table. I
248
CHAPTER 5 Exponential and Logarithmic Functions
Figure 3 provides a second illustration of the definition. Here x is the input to
the function g, yielding g1x2. Then g1x2 is the input to the function f, yielding
f1g1x22. Notice that the inside functio
158
CHAPTER 3 Linear and Quadratic Functions
3.5 Assess Your Understanding
Are You Prepared?
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
2. Write the interval 1  2, 7] using inequality notatio
SECTION 5.4 Logarithmic Functions
(b) Graph f1x2 = cosh x using a graphing utility.
(c) Refer to Problem 122. Show that, for every x,
Problems 122 and 123 provide definitions for two other transcendental functions.
1cosh x22  1sinh x22 = 1
122. The hyper
268
CHAPTER 5 Exponential and Logarithmic Functions
For example, take the irrational number p = 3.14159 . Then an approximation
to ap is
ap L a3.14
where the digits after the hundredths position have been removed from the value
for p. A better approximati
SECTION 5.3 Exponential Functions
EXAMPLE 4
Graphing an Exponential Function
1 x
f1x2 = a b
2
Graph the exponential function:
1  10
a b
= 1024
2
3
1 3
= 8
a b
2
2
1 2
a b
= 4
2
1
1 1
a b
= 2
2
Figure 22
y
6
ols
10
To
1 x
f(x) a b
2
(2, 4)
0
1
1
1 1
a
CHAPTER 5 Exponential and Logarithmic Functions
EXAMPLE 2
Using Properties (1) and (2)
(a) 2log2 p = p
(b) log0.2 0.22 =  22
2
Now Work
(c) ln ekt = kt
15
PROBLEM
Other useful properties of logarithms are given next.
Properties of Logarithms
In the foll
288
CHAPTER 5 Exponential and Logarithmic Functions
(d) To find f1, begin with y =  ln1x  22. The inverse function is defined
(implicitly) by the equation
x =  ln1y  22
Proceed to solve for y.
 x = ln1y  22 Isolate the logarithm.
ex = y  2
Change
264
CHAPTER 5 Exponential and Logarithmic Functions
In Problems 1924, the graph of a function f is given. Use the horizontalline test to determine whether f is onetoone.
19.
20.
y
3
3
3
3 x
3 x
y
3
3
y
y
3
23.
3 x
3
3
3
22.
21.
y
3
y
3
24.
2
3
3
3
3
3
258
CHAPTER 5 Exponential and Logarithmic Functions
receives as input Indiana and outputs 6.2 million. So the inverse receives as input
6.2 million and outputs Indiana. The inverse function is shown next.
Population
(in millions)
State
6.2
Indiana
6.1
Was
SECTION 5.5 Properties of Logarithms
303
5.5 Assess Your Understanding
Concepts and Vocabulary
1. loga 1 =
8. If loga x = loga 6, then x =
log5 7
, then M =
9. If log8 M =
log5 8
2. loga a =
3. alogaM =
4. loga ar =
5. loga(MN) =
+
M
b =
N

6. loga a
7.
The Chapter Test Prep Videos are stepbystep test solutions available in the
Video Resources DVD, in
, or on this texts
Channel. Flip
back to the Student Resources page to see the exact web address for this
texts YouTube channel.
1. Graph f1x2 = 1x  324
SECTION 4.1 Polynomial Functions and Models
173
(b) The two xintercepts divide the xaxis into three intervals:
10, 22
1  q , 02
12, q 2
Since the graph of f crosses or touches the xaxis only at x = 0 and x = 2, it
follows that the graph of f is either
CHAPTER 4 Polynomial and Rational Functions
Examples of power functions are
f1x2 = 3x
f1x2 =  5x2
f1x2 = 8x3
f1x2 =  5x4
degree 1
degree 2
degree 3
degree 4
rit
er
an
d
To
ols
The graph of a power function of degree 1, f1x2 = ax, is a straight line, wit