1 PRECALCULUS

REVIEW

1.1 Real Numbers, Functions, and Graphs

= . , = . ... = .

π= . ...

repeating
periodic
π

number
∈

a∈ a

Additional properties of real numbers are

discussed in Appendix B. Zahl = {. . . , − , − , , , , . . . }
, , ,....
a a p/q p
q q= √
π a a

= . ..., = . = . ..., = . = . ...

− −
FIGURE 1

1
2 CHAPTER 1 PRECALCULUS REVIEW

|a| a va a |a|
a a a≥
FIGURE 2 |a| a |a| = distance from the origin =
−a a<

| . |= . |− . |= .

|a| = |−a|, |ab| = |a| |b|

|b − a| a a b |b − a|
a b
− − a b
a b |b − a|
FIGURE 3 a b
if the decimal
|b − a|
expansions of a and b agree to k places (to the right of the decimal point), then the
distance |b − a| is at most −k a= . b= .
− a b
| . − . |= .
|a + b| |a| + |b| a b
a b
a+b |a + b| < |a| + |b| | + |=| |+| | |− + | =
|− | + | | = |a + b| |a| + |b|
a a

|a + b| ≤ |a| + |b|

a<b
a b b−a

a b a b a b a b
FIGURE 4 v a b v a b v a b v a b
a b u u

va [a, b] x a≤x≤b

[a, b] = {x ∈ : a ≤ x ≤ b}

{x : a ≤ x ≤ b} x
a va

(a, b) = {x : a < x < b} , [a, b) = {x : a ≤ x < b}, (a, b] = {x : a < x ≤ b}

(−∞, ∞)

[a, ∞) = {x : a ≤ x < ∞}, (−∞, b] = {x : −∞ < x ≤ b}

a b
FIGURE 5 a ∞ −∞ b
 S E C T I O N 1.1 Real Numbers, Functions, and Graphs 3

|x| r
(−r, r) |x| < r
−r r
FIGURE 6
|x| < r ⇔ −r < x < r ⇔ x ∈ (−r, r)
(−r, r) = {x : |x| < r}

r r |x − c| < r ⇔ c−r <x <c+r ⇔ x ∈ (c − r, c + r)

c−r c c+r
< ≤ r a c
FIGURE 7 (a, b) = (c − r, c + r)
(a, b) [a, b] c = (a + b)
a+b b−a r = (b − a)
c= , r=

EXAMPLE 1 [ , ]
[ , ] c= ( + )=
r= ( − )=
FIGURE 8 [ , ] [ , ]= x∈ : |x − |≤
|x − |≤
EXAMPLE 2 S= x: x− >
x− ≤
In Example 2 we use the notation ∪ to
denote “union”: The union A ∪ B of sets x− ≤ ⇔ − ≤ x− ≤
A and B consists of all elements that
belong to either A or B (or to both).
− ≤ x≤ ( )

− ≤x≤ ( )

−
x− ≤ x [− , ] S complement
x not in [− , ] S
FIGURE 9 S= x: x− >
S = (−∞, − ) ∪ ( , ∞)

Graphing

The term “Cartesian” refers to the French x y (a, b) P

philosopher and mathematician René x a
Descartes (1596–1650), whose Latin y b a b x y a P
name was Cartesius. He is credited (along x y
with Pierre de Fermat) with the invention of
( , )
analytic geometry. In his great work La
Géométrie, Descartes used the letters
x, y, z for unknowns and a, b, c for
(x, y)
constants, a convention that has been x< y<
followed ever since. d P = (x , y ) P = (x , y )
P P
a = |x − x | b = |y − y |
d = a + b = (x − x ) + (y − y )
4 CHAPTER 1 PRECALCULUS REVIEW

y y

P= a b
b
− + + +

x x
− − a
− V
− − + −
−

FIGURE 10

y
Distance Formula P = (x , y ) P = (x , y )
P = x y
y

d d= (x − x ) + (y − y )
|y −y |
y P = x y
|x −x |
r
x (a, b) (x, y) (x, y)
x x
(a, b) r
FIGURE 11 d
(x − a) + (y − b) = r
y

x, y
r (x − a) + (y − b) = r
b
a, b

DEFINITION f D Y
x x D y = f (x) Y
a
FIGURE 12
f :D→Y
(x − a) + (y − b) = r
D a f x ∈ D f (x)
va f x a R f Y
f (x)

R = {y ∈ Y : f (x) = y x ∈ D}

A function f : D → Y is also called a f y x

“map.” The sets D and Y can be arbitrary. D
For example, we can define a map from the
set of living people to the set of whole
numbers by mapping each person to his or
her year of birth. The range of this map is x f f x
the set of years in which a living person x f f x
was born. In multivariable calculus, the D Y u u u
domain might be a set of points in FIGURE 13 FIGURE 14 f
three-dimensional space and the range a f (x) Y x∈D x
set of numbers, points, or vectors. f (x)
 S E C T I O N 1.1 Real Numbers, Functions, and Graphs 5

numerical f
f
f (x) x va a
D y = f (x) y va a
x
f √ x
√ f (x) = −x
D = {x : x ≤ } −x −x ≥

f (x) a D a R

x {y : y ≥ }
x {y : − ≤ y ≤ }

{x : x = − } {y : y = }
x+

y
y=f x
a y = f (x) (a, f (a)) a
D x=a x
f a a f a y f (a) |f (a)|
f x x
x f (x) c f (c) =
a c
x x
FIGURE 15

EXAMPLE 3 f (x) = x − x

x − x = x(x − ) = .
√
f (x) x= x=±

TABLE 1

x x − x −
x
− − − − −
−
−
−
FIGURE 16 f (x) = x − x

W t
6 CHAPTER 1 PRECALCULUS REVIEW

w g W g y

TABLE 2

t W t W

x
−
−
−

g t y
FIGURE 18 y −x =
FIGURE 17

y x
y −x = (x, y)
x
y x= y=±
a
x=a

f (x)

• a (a, b) f (x ) < f (x ) x , x ∈ (a, b) x <x

• a (a, b) f (x ) > f (x ) x , x ∈ (a, b) x <x

f (x)
x
f (x) a f (x ) ≤ f (x ) x <x ≤
< a

y y y y

x x x x
a b

g g g a b g u
u g g
v yw
FIGURE 19

• f (x) v f (−x) = f (x)

• f (x) f (−x) = −f (x)
 S E C T I O N 1.1 Real Numbers, Functions, and Graphs 7

• v y P = (a, b)
Q = (−a, b)
•

P = (a, b) Q = (−a, −b)

y
y
y
−a b b a b
b a b
−a
x x x
−a a a
−a −b −b

v u : f −x = f x u : f −x = −f x v
y y
u y u g
FIGURE 20

EXAMPLE 4
a f (x) = x g(x) = x − h(x) = x + x

a f (−x) = (−x) = x f (x) = f (−x) f (x)

g(−x) = (−x)− = −x − g(−x) = −g(x) g(x)
h(−x) = (−x) + (−x) = x − x h(−x) h(x)
−h(x) = −x − x h(x)

E X A M P L E 5 Using Symmetry f (x) =

x +
f (x) f (x) > f (−x) = f (x)
x y
f (x) x≥ x

x f (x)
|x|

TABLE 3

x y
x +
f x =
x +

±
± x
− −
FIGURE 21
8 CHAPTER 1 PRECALCULUS REVIEW

a a a

DEFINITION Translation (Shifting)

Remember that f (x) + c and f (x + c) • a a a y = f (x) + c |c| vertically
are different. The graph of y = f (x) + c c> c c<
is a vertical translation and y = f (x + c) • a a a y = f (x + c) |c| horizontally
a horizontal translation of the graph of c< c c>
y = f (x).

f (x) = /(x + )

y u y y
u w u

x x x
− − − − − − −

y=f x = y=f x + = + y=f x+ =

x + x + x+ +
FIGURE 22

EXAMPLE 6 f (x) = x

y y

x x
− − − −
− −

f x =x
FIGURE 23

( , ) f (x)
( ,− ) g(x) = (x − ) −

y a a
y=f x

x DEFINITION Scaling
• a a
y = kf (x) k>
− k <k<
k k< x
− • a a y = f (kx) k>
y=− f x
<k< k<
FIGURE 24 y
k=−
 S E C T I O N 1.1 Real Numbers, Functions, and Graphs 9

a
|k|

Remember that kf (x) and f (kx) are EXAMPLE 7 f (x) = (π x) f ( x) f (x)

different. The graph of y = kf (x) is a
vertical scaling, and y = f (kx) a f (x) = (πx)
horizontal scaling, of the graph of
y = f (x).
• f ( x) = ( πx) y = f (x)

• y = f (x) = (π x) y = f (x)

y y

x x x

− − −
y y −

FIGURE 25 y=f x = πx : V :
f (x) = (πx) y=f x = πx y= f x = πx

1.1 SUMMARY

a a≥
• |a| =
−a a<
• |a + b| ≤ |a| + |b|
• a b

(a, b), [a, b], [a, b), (a, b]

(a, b) = {x : |x − c| < r}, [a, b] = {x : |x − c| ≤ r}

c = (a + b) r = (b − a)
• d (x , y ) (x , y )

d= (x − x ) + (y − y )

• r (a, b)

(x − a) + (y − b) = r
• zero root f (x) c f (c) =
10 CHAPTER 1 PRECALCULUS REVIEW

x=a
f (x ) < f (x ) x <x
f (x ) ≤ f (x ) x <x
•
f (x ) > f (x ) x <x
f (x ) ≥ f (x ) x <x
• f (−x) = f (x) y
• f (−x) = −f (x)
• f (x)
f (x) + c |c| c> c<
f (x + c) |c| c< c>
kf (x) k
k< x
f (kx) k k>
k< y

1.1 EXERCISES
Preliminary Questions
a b a<b |a| > |b| a ( , ) (− , ) ( ,− ) (− , − )
|a| = a |a| = −a
|−a| = a (x − ) + (y − ) =
a b
|a + b| < |a| + |b| f (x) =
a f

x= y=− f

f (−x) = −f (x)

Exercises
r In Exercises 13–18, express the set of numbers x satisfying the given
|r − π | < − condition as an interval.
|x| < |x| ≤
a=− b=
a a<b |a| < |b| ab > |x − | < |x + | <

a< b − a<− b < | x− |≤ | x+ |<

a b
In Exercises 19–22, describe the set as a union of finite or infinite in-
In Exercises 3–8, express the interval in terms of an inequality involving tervals.
absolute value.
{x : |x − | > } {x : | x + | > }
[− , ] (− , ) ( , )
{x : |x − | > } {x : |x + x| > }
[− , ] [ , ] (− , )

In Exercises 9–12, write the inequality in the form a < x < b. a a> |a − | <
|x| < |x − |<
a− < |a| >
| x+ |< | x− |< |a − | < ≤a≤
 S E C T I O N 1.1 Real Numbers, Functions, and Graphs 11

a
a
f : {r, s, t, u} → {A, B, C, D, E}
a
v a f (r) = A f (s) = B f (t) = B f (u) = E

v a D
v a − R
D R
x
x: < In Exercises 41–48, find the domain and range of the function.
x+
f (x) = −x g(t) = t
{x : x + x < } Hint: y=x + √
x− f (x) = x g(t) = −t

|x − | = |x − | + f (x) = |x| h(s) =

s

a>b b− > a − a b f (x) = g(t) =

x t
a> b<
In Exercises 49–52, determine where f (x) is increasing.
x |x − | < |x − | <
f (x) = |x + | f (x) = x
|a − | < |b − | <
|(a + b) − | < Hint: f (x) = x f (x) =
x +x +
|x − | ≤ In Exercises 53–58, find the zeros of f (x) and sketch its graph by plot-
a |x + | ting points. Use symmetry and increase/decrease information where
appropriate.
|x − |≤
f (x) = x − f (x) = x −
|a − | ≤ |b| ≤
a |a + b| f (x) = x − x f (x) = x
|a + b|
f (x) = −x f (x) =
(x − ) +
|x| − |y| ≤ |x − y| Hint:
y x −y

r = . Hint: r −r y y
r = . ...

/ / x
x
If the decimal expansions of numbers a and b agree
to k places, then |a − b| ≤ −k
k a b do not agree
at all |a − b| ≤ −k y y

x x
a ( , ) ( , ) ( , ) ( , )
( , ) (− , ) (− , − ) (− , )

( , ) FIGURE 26
a r=
( ,− )
a f (x) = x g(t) = t − t

F (t) =
( , ) t +t