Chapter O Preliminaries

Lecturer: Yuanpeng Zhu

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Calculus (I)!
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df(x) N0-4ac N 62-4a0
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x+ +
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Recall The Concept of Function

Input x Output f(x)

Function f

x• • f(x)
a• bo • f(a) = f(b)

Domain Range

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The Definition of Function

egnR
a
4
3

Domain
1 ！
1-

DEFINITION Afunction f from a set Dot a set Ysi a rule that assigns a unique
value f(x) in Yto each x in D

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Two Important Symbols and The Definition of Function
1 V For any
2 3 Exist

> Definition of function (Page 29)

Given two number sets A and B, fi the map ffrom A to B satisfies that
Vx E A, o n e and only one y e B, such that y = f ( x ) the dependent variable

then, we call the map f as a function from Aot B.

the independent variable

Domain Range 5/60

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Check which one is function

Two y values for one value Two times for one temperature
of x fails test—not a function a—function

Temperature
Vertical Line Test
A graph represents a function if and only if ti passes the vertical line test: Every
vertical line intersects the graph at most once. A graph that fails this test does not
represent a function.
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Function

• f(x) read "fofx" or "fat x", denotes the value that f assigns ot .x
Example 1

For f(x) =x2-2x, find and simplify {(4+h) - f(4)

f(x + h) y = f(x)

f(x + )h - f(x)

f(x) -
Soluti on:

f(4+h) - f _oh oh+3 - = 6 + h

For h > 0

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Natural
d o m a i no f Funocitn

• When no domain is specified for a function, ti is called the

Natural domain.
Example 2
Find the natural domains for
1
(a) f ( x ) =• {XER: x# 3}
x-3
(b)g(t) = V9- [-3,3]
1
(c)h(w) = .
(-3,3)
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Some examples of functions

Function Domain (x) Range (y)

？ (-00, 00) [0, 00)
y = 1/x (-00, 0) U (0, 00) (-00, 0) U (0, 00)
y = Vx [0, 00) [0, 00)
y = V4 x- (-00,4] [0, 00)
=y VI-x2 [-1, 1] [0, 1]

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The graphics of functions

If (x, y) lies on the graph of

f, then the value y = f(x) si the height of
the graph above the point x(or below xif
f(x) is negative).
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The graphics of functions

(-2, 4) 41
(2, 4)
2- 4 y = 2x
3
1-
0 0 2- 2( %
)
1 1 (-1, 1) a • (1,1)
914
3 12

→X
-1 1 2
2 4 Graph of the function
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The graphics of functions

Pressure
P (pressure uPa)
Time Pressure Tm
ie
0.00091
0.00108
-0.080
0.200
0.00362
0.00379
02.17
0.480
01. - • Data
0.00125 0.480 0.00398 06.81
0.00144 06.93 0.00416 08.10
00.0162 08.16 0.00435 08.27
00.0180 0.844 00.0453 07.49
00.0198 0. 00.0471 05.81 -0 .2 -
00.01 00.02 0.003/ 0.004 00.05 00,06 →t (sec)

00.0216 0.603 0.00489 0.346 -04. -

0.00234 0.368 00.0507 00.77
00.0253 0.099 00.0525 -01.64 -06. -
0.00271 -0.141 00.0543 -03.20
0.00289 -0.309 0.00562 -0.354
00.0307 -0.348 00.0579 -02.48 Asmooth curve through the plotted points
00.0325 -0.248 0.00598 -00.35 gives a graph of the pressure function represented by the
0.00344 -0.041
accompanying tabled data

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The graphics of functions

Graph of a catenary or
hanging cable. (The Latin word catena
means "chain.") 13/60
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多
The graphics of functions

A
Y
y = x- y = f(x)
-3

1
y= 1
2 ifx<1
g(x) = 0 if1 < x < 2
1- fi x > 2 2- 0 2 →x

-1 3
To graph the
- |1
function y= f(x) shown here,
we apply different formulas ot
different parts of its domain
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The graphics of functions

У 11 = 50
8

Area = A(5)
Aera = A2()
5
-4
3 02
A 81
16 i f > 3
8-
41
y= J y=/( 12
5 6 7 8 01
The graphics of some important functions

Range: y>0

y = loga x
y = log3x

→x
y= logs.x

y = log10x
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The graphics of functions

Shift Formulas
Vertical Shifts
y = f(x) + k Shifts the graph of fup kunits if k > 0
Shifts ti down |k| units if k < 0
Horizontal Shifts
y= f(x + h) Shifts hte graph of fleft hunits fi h > 0
Shifts ti right |h| units if h < 0

y=
/ 2x + 2 Add a positive Add a negative
= +1 constant to .x constant to .x

2
2
（* +3 ） = x y =（
* 2）
~

1 unit -

→
2-
2 units 3- →x
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The graphics of functions

inan

У
y = /x|
y = - x 4 y=/x-2|-1
2
1

-2 1- 1 2 3 >x - 4 2-
- 1
4 6 →x

FIGURE The graph of y = x/

shifted 2 units to the right and 1 unit
down
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 The graphics of functions
Vertical and Horizontal Scaling and Reflecting Formulas

For c > ,1 the graph is scaled:

y = cf(x) Stretches the graph of f vertically by a factor of .c

=y df(x). Compresses the graph of f vertically by a factor of c.

y= f(cx) Compresses the graph of f horizontally by a factor of c.

y = f(x/c) Stretches the graph of f horizontally by a factor of c.
For c = -1, the graph is reflected:
y = - f(x) Reflects the graph of f across the x-axis.
y = f（- x ） Reflects the graph of f across the y-axis.
y= V-x
= 3Vx ٧=V٤
y= V3x
Stretch y= Vx compress

→stretch
y=V* 3 -2 4
compres fy=Vx y = Vx/3 -1|

2 3 Vx

FIGURE Vertically stretching and FIGURE Horizontally stretching and FIGURE Reflections of the graph
compressing the graph y = Vx by a
factor of 3 riyb afactor fo
compressing hte graph y= V y = Vr across hte coordinate axes
The graphics of functions
Vertical stretch or compression; •Vertical shift
reflection about sxai- if negative

y = af(b(x + c)) + d
Horizontal stretch or compression; Horizontal shift
reflection aboutsayx-i if negative
- 1 ）
- 2

The graph fo y = f(x - )b is the graph For c >0, the graph of y = cf(x) si the graph For c< Q,the phagr of y= ef(x) ishet graph
of roF a < 0, the graph of y = f(ax) si the graph of y = x()f
of
y = f(x) shifted horizontally yb b units of y = f(x) sealed vertically by a factor of c y = f(x) scaled vertically yb a factor of k and scaled horizontally yb a factor of(a)and reflected acors the
(right fi b > 0and left fi b < .)0
(broadened fi 0 < c < 1 and steepened fi c > I). reflected across hte x-axsi (broadened fi ya'xis (broadened fi - 1< a< 0 nda steepened fi a< - 1).
y = f(x)
（
- 2x）

= f(x)
（x 2）

3/x)
—xY y = - ( 0)

y = 2/(x)
/(x)
2)-y=
({
y= 2x-(f)
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The graphics of functions

S U M M A RY Transformations

Given the real numbers a, b, c, and d and the function f, the graph of
y = cf(ax - b)) + d si obtained from the graph of y = f(x) ni the
following steps.
horizontal scaling
by a factor o f a
y = f(x) →y = f(ax)
horizontal shift
by b units
→y = f(ax - b))
vertical scaling
by a factor of cl
→y = cf(ax - b))
vertical shift
by d units
→y = cf(ax - b)) + d

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The graphics of functions

Step 1: Horizontal scaling

y= 2x + 1|
У

Basic curve

2- ½
-1

Step 2:Horizontal shift

=2 x ＋
1］
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Monotonic Functions

DEFINITIONS Let f be a function defined on na interval / and let x, and 2x be

two distinct points in .I
1. If f(x2) > f(x) whenever x, < xz, then f si said ot be increasing on I.
2. fI f(x2) < f(x) whenever x, < xz, then f si said ot be decreasing on I.

h h = 30t - 517
Downward
- 40-
ground (meters)

path of
Height above

30 the stone

Upward 20
path of
the stone 10

0 5 6
m
Tie (seconds) 23/60
 Even a n d Odd Functions
If the domain of the function f(x) is symmetric with respect to 0, and

Vx, f(-x) = f(x), （ y）

Symmetry (-x, y)
about y-axis

→f(x) is an Even function.

The graphs of even function si symmetre with respect ot y - axis

Symmetry
about origin
/（
.y）
V x, f （
- x）
= / （
x），
(-x, y-)
→ f(x) is an Odd Function
(b)

The graphs of odd function si symmetre with respect ot the origin of coordinate (0,0)
Even a n d Odd Functions

Odd function—if (x, y) is on the No symmetry-neither an

graph, then (-x, -y) si on the graph. even nor odd function.
Even function-if (x, y) si on the
graph, then (-x, y) si on the graph.
20-

20
( - 0 . 5 . 2.67)
01 -
10-
(1.5, 0.53)
_⺾

wta
(-1.5. -0.53)
- 10-
(-2, - 12) (2, - 12) 1
(0.5, - 2.67)
⼀ 10 ⼀ y= x - +x3

y =x*
- 2x2 - 20
3-0-
02- -

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Inverse F u n c t i o n s
DEFINITION Inverse Function
Given a function ,f its inverse (if ti exists) si a function f*' such that whenever
y= f(x), then f'( y) = r
x is in the domain of fand
* =f'(y) si ni hte range off! Two values of x
correspond to y.
m
faps r toy (x. y)
y =/(x)
x(. y)
• y = f(x)

ysi ni hte domain of f ' and

y= f(x) is ni the range of .f maps y to.x

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Inverse F u n c t i o n s
DEFN
IT
IO
IN One-to-One Functions and the Horizontal Line Test
A function f is one-to-one on a domain D fi each value of f(x) corresponds to exactly
one value of xni D. More precisely, f si one-to-one on D if f(x,) # f(x2) whenever
x, # X2, for x, and zx ni D. The horizontal line test says that every horizontal line
intersects the graph of a one-to-one function at most once
One-to-one function:
Each value of y ni the range Not one-to-one function:
corresponds ot exactly Some values of y correspond
one value fo .x to more than one value of.x.

Y
A

The Horizontal enL

i Test for One-to-One Functions
A function y= f(x) si one-to-one if and only if its graph intersects each hori-
zontal line at most once.

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Inverse Functions

f(x) = -r is not f(x) = 12si 1-1 (x) =risl-

I-1 on (-00, co). on (-00, O]. n [0. 00

ffails hte
horizontal
line test.

fis one-to-one fis one-to-one

1 1 17 4x and has an inverse
on (-00, 0].
and has na inverse
on [0, 0o.)

Domain: ( - 0 0 , 0o) Domain: ( - 0 , ]0 Domain: [0, co)

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 Inverse Functions

DOMA
N
I OFf RANGE FO f-'
a)( To find the value of fat x,w e start at ,x (b) The graph of f-' si hte gaprh of f.but
(e)oT darw the garph of f"' in the )d( nehT ew interchange eht eltxrs. dna y.
go up ot the curve, and then over ot the y-axis. with x and y interchanged. oT find the x that more usual way, we reflect the eW now evah a norma-looknig garph off
gave y, we start at y and go over to the curve
and down to hte x-axis. The domain of f i s hte
range off. The range of f ' si het domain of .f
FIGURE The graph ofy = f*'(x) is obtained yb reflecting the graph ofy = f(x)
about the nile y = .x

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Inverse Functions

Symmetry about y = x means...

У
y = f(x) /

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Inverse F u n c t i o n s
PROCEDURE Finding an Inverse Function
Suppose f is one-to-one on an interval I. To find f ' :
.1 Solve y = f(x) for x. fI necessary, choose hte function that corresponds ot .I
2. Interchange xand y and write y= f(x).

y= f(x) = * 1- averse function

r f(x) = * -

y=f*'(x) = Vx+ 1+
V
x

↓— ⼗

+ y =5'(x) = - Vx+1
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Inverse F u n c t i o n s

The function f(x) = x2 + 6 and its

The function f(x) = Vx - 1x( ≥ 1)
inverse f(x) = ½- 3are and its inverse f(x) = 2x + 1(r≥ 0)
symmetric about the line y= .x
are symmetric about y = x.
f'(x) = 21 + 1(x ≥0)
y= x

y = X

f(x) = 2x + 6

5(0)=-3 f(x) = Vx - 1(x≥ 1)

+

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Inverse F u n c t i o n s
DEFINITION Logarithmic Function Base b
For any base b > 0, with b # 1, the logarithmic function base b, denoted y = log, x,
is the inverse of the exponential function y= b*. The inverse of the natural exponential
function with base b = e si the natural logarithm function, denoted y = In x.
y = e*
Graphs of b* and log, x are
symmetric about y = x.

= 、b ＞ 1 y = 2*
5

y= x
y =x
eo< (1, e)
y = In x
(O, 1) y = log, * y = log2x

2 > x 4
(, )0

(b)

(a)
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Inverse Functions

nI x= y → e = .x

nI particular, because e' = e, we obtain

In e = 1.

THEOREM -Algebraic Properties of hte Natural Logarithm For any numbers

b > 0 and x > 0, the natural logarithm satisfies the following rules:
1. Product Rule: nI bx = Inb + Inx

2. Quotient Rule: Inf =nIb- nIx

.3 Reciprocal Rule: Rule 2w i t h b = 1|

4. Power Rule: In x' = r i n x

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Inverse Functions

Inverse Properties for a and loga x

1. Base a: alog* = x, loga a* = x, a > 0, a # 1, x > 0
2. Base e: enx = x, In e* = x, x > 0

Every exponential function is a power of the natural exponential function.

a* = exina
That is, a* is the same as e" raised to the power In a: a* = ek* for k = Ina.

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Inverse F u n c t i o n s

Change-of-Base Rules
Let b be a positive real number with b * .1 Then
In x
b* = e*Inb, for all x and 10g, x = nI b' forx > 0.
More generally, fi c is a positive real number with c # ,1 then

b* = crlogb, for al x and logs

log,xx =
=
10ge b forx >.0

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Inverse F u n c t i o n s
THEOREM Existence of Inverse Functions
Let f be a one-to-one function on a domain D with a range R
. Then f has a unique
inverse f ' with domain Rand range D such that
f'(f(x)) = x and f f ' (y)) = ,y
where x is in D and y is in R.

Inverse Relations for Exponential and Logarithmic Functions

For any base b > 0, with b # ,1 the following inverse relations hold:
11. blogor = ,x for x > 0
12. logob* = x, for real values of x

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Operations on Functions

Function Formula Domain

f+ g (f + g)(x) = V
x + VI -x [0, 1] = D(f) ND(g) y=f+ g
f- g (f - g)(x) = Vx - Vi x- [0, 1] 8(x) = V1 - x f(x) = Vx
g- f (g - /)(x) =V1 - x- Vx [0, 1]
f•g (f •g)(x) = f(x)g(x) = Vx(1 - x) [0, 1]
f/g 6(3) - 282
(3)) VE
I* [0, 1) x( = 1excluder
y = g

8/f 8((xx))
(x) = 7 =V (0, 1] (x = 0 excluder
> x

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Composite Functions

Function Function
8
→ I = g(r) → →y = f u ) = f(g(x))

(a)

Domain of g Range of g Domain off Range of f o g

g

• f(g(x,)

8(12) si outside domain g(r,) si ni domain

of ,f so.r, si not ni of .f os x, si ni
domain of fo g. domain of fog.
(b)
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Composite Functions
DEFINITION fI fand g are functions, the composite function f • g ("f com-
posed with g") si defined by
(f ° g)(x) = f(g(x)).
The domain of f• g consists of hte numbers xni hte domain of g for which g(x)
lies ni the domain of f
fog

D
o f(g(x))

x → 8 g(x) f → f(g(x))

Acomposite function fog uses

8(x) the output g(x) of the first function gas the input
for the second function f.
Arrow diagram for fo g. I f lies ni the
domain of g and g(x) lies ni the domain of f, then the
functions f and gcan be composed ot form (f ° g)(x).
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Composite Functions
Composition of g with f is defined as follows
(gof)(x) = g ( f ( x ) )
the domain of g o f is
(I) x is in the domain o f g
(2) f(x) is in the domain of g
C
8

y=f(x)
* g(V)=g((x))=gof(x)
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Composite Functions 多

У
9

‫اﻟﺴﻨﺎ‬
8

f (g(5)) = ?
7

6
5

g(f (5)) ?=
2 8（
1）

⽞
17
9

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Trigonometric Functions and Their Inverses

s = r0 ( in radians)
©

On a circle of radius r. On a circle of radius 1,

radian measure of 0 i s radian measure of 0 is .s

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Trigonome tric Functions and Their Inverses

Terminal ray>
Initial ray
→x
Positive Initial ray
Negative
measure Terminal measure
→x
ray y

Angles ni standard position ni the xy-plane.

砍- →x

Nonzero radian measures can be positive or negative and can go beyond 2mr.
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Trigonometric Functions and Their Inverses

(O.)1
（-⼩吆） (6-2) 43()
(- . )
( -; ) (4.23)

°021

45 *m/3

/4
=

m
6-0
( - . ) 150P ( . ))

3=/2
1-3

m°
-

357°
伊- 脂 5=

/4
/6
18 0°m a 0 ° = 0 radians
(-I.0)Ф • (10.)|
360° - 2=

/4
- 617
0- 噌 3 30

Sw
210° °-
11.

3=/2|
-2-1 66 7(7-1

315° 0S°#/3
3A
-
/

%
°52

7
3-0
240° -

A4
( .- ) (½. - )

270° -
(4-2) (½. - )

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Trigonometric Functions and Their Inverses

У
sin 0 = So 0= ,
C tan 0 =

hypotenuse
opposite cot 0 =
P(r, y)

adjacent
sin 0= OP
hyp
csc 0=hyp x
opp
cos @=ahydpi sec o= hadj
yp
Opp
tan 0= adj cot o= aOPP
dj
FIGURE Trigonometric Apositive angle 0 results from
a counterclock wsie rotation.
ratios of an acute angle.
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Trigonometric Functions and Their Inverses

The graphs of y= sin 0 and its reciprocal, y= csc 0 The graphs of y= cos 0 and its reciprocal, y= sec 0

= s e c0
y = c s c0

= sin 0
y = cos 0

3m Mm 2 3 тЛт 4
3？

2
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Trigonometric Functions and Their Inverses

The graph of y= tan @has period m

.r The graph of y = cot 0 has period r.

y = tan 0 y = cot 0

Вт 2 m
S 3т 4m

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Trigonometric Functions and Their Inverses

y = tan x

D om ai n： - ∞ < x ＜8
Range: -1sys1
Domain: — 00 < x < 00|
Range: - 1s y s 1
Domain: x*场 ⼟贺
Range: - 0 < y<00
Period: 2 Period: 2
Period:
(a) (b) (c)

=C
S
Cx

2m
-

Domain: 3*± , ± . ... D om ai n: x¥ 0，⼟司、⼟2 . . . .

Range: y s - l o r y z 1
Domain: x * 0，⼟ 、⼟2. ，
Range: - ∞ < y < 8
. ..
Range: y s - 1 or y≥1 Period: 2 Period:
Period: 2
(d) (e) (f)

FIGURE Graphs of the six basic trigonometric functions using radian measure. The shading for each 48/60
trigonometric function indicates its periodicity.
Trigonometric Functions and Their Inverses
Trigonometric Identities
Periods of Trigonometric Functions Reciprocal Identities
Period n : tan (x + mr) = tanx COs 0
cot (x + m) = cotx atn ®= sni 0 cot 0 =
1
＝
Cos e tan 0 sin 0
Period 2m: sin (x + 2mr) = sin x 1 1
cos (x + 2m) = cos x C
S
C 0= sec 0 =
sin 0 COS 0
sec (x + 2m) = sec x
csc (x + 2m) = esc x Pythagorean Identities
sin? 0 + cos?0 = 1 1 + cot 0 = csc? 0 tan 2e + 1 = s ec ' e

Double- and Half-Angle Formulas

Even Odd sin 20 = 2sin 0 cos 0 cos 02 = cos? 0 - sin? 0
cos ( x ) = cos.x sin (-x) = -sinx 1 + cos 20 1 - cos 02
sec (-x) = sec.x tan (-x) = -tanx cos' 0 = - 2 sin? 0 =
2
csc (-x) = -csc x
cot (-x) = - cotx Addition Formulas
cos (4 + B) = cos Acos B - sin 4 sin B
sin A
( + B) = sin Acos B + cos Asin B
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Trigonometric Functions and Their Inverses

Domain restrictions that make the trigonometric functions one-to-one
tan x

sin x
cosx

y = sinx y = COS x y = tan x

D omain: [ - / 2, 7/ 2] o m ain : ( 0, D omai n: ( - / 2, m /2)
Range: (-1,1]| ange: 1-[, 1 Range: (-0o, co)

csex

y = cot x y = seex y = csex

Domain: (0, m) D omain: [ 0, / 2)U ( m/ 2, m) Domain :[ - m/ 2, 0) U (0. m/ 2 ]
Range: (-00, co) Range: (-00, - 1] U [1,∞0) Range: (-00, -1]U ,1[ ∞0)
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Trigonometric Functions and Their Inverses

Restrict hte domain of y= sin xot | - ₴ J Restrict the domain of y = cos x to [0, m].

y= sin.x
y = cos x

Range =—[ ,1 1]|

Range = [-1, 1:
2

1
Domain = | - ₴
D om ai n = [0, m)

DEFINITION Inverse Sine a n d Cosine

y = sin' xsi hte value of y such that x= sni ,y where - / 2 ≤ y ≤ 1/2.

y = cos' xsi the value of y such that x = cos ,y where 0 ≤ y ≤ .
The domain of both sin ' xand cos 'xis {x: {x:-1 ≤ x ≤ 1}.
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Trigonometric Functions and Their Inverses

The graphs of y= sin xand

y= sin' rare symmetric The graphs of y = cos x and
about the line y = x.
y= cos ' erar symmetric
about the line y = .x
⽇. y
= s i n
2 =x y x=
y= c o s
Range of sin.x
= Domain of sni x' Range of cos x
= Domain of cos"x
—⼗ [-1, 1] y = cos x
1| [-1, 11
2
1-

y = sin.

[0 . m ]

Restricted domain of cos v

= Range of cos -' x
Restricted domain of sin x
= Range of s i n x
52/60
Trigonometric Functions and Their Inverses
arcsin x arccos x

sin @= * >= V3/2 тт/3 TT/6

(x, y)
V2/2 =/ 4
Cos 0 = x
0= c o s ' x 1/2 тт / 6 п/3
- 1/2 - 7/ 6 2 / 3
- V2/2 - /4 3 / 4
- V3/2 - тт/3 5т / 6

arccos x
Taercos )2 -72
V3 T
arcsin x
→ x

FIGURE arcsinx and arccosx are

complementary angles (so their sum si m/ 2). sin - V
3 C
so(5) 1-=
(a) (b) 53/60
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Trigonometric Functions and Their Inverses

DEFINITION Other Inverse Trigonometric Functions

y = tan' xsi the value of ysuch that x = tan ,y where - 1/2 < y < T/2.
y= cot ' xsi the value of ysuch that x= cot ,y where 0 < y < .
The domain of both tan 'x and cot' xsi {x:-0 < x < 00}.
y = sec x si the value of y such that x = sec ,y where 0 ≤ y ≤ ,i with y # +/2.
y= esc r si the value of ysuch that x= csc ,y where - / 2 ≤ y ≤ m/2, with
y + 0.
The domain of both sec ' rand c s xsi {x:|x|≥1}.

54/60
Trigonometric Functions and Their Inverses

Restricted domain of tan x

y = nat x
si ( - ₴)

y= c o t 'x
Range of tan x' Range of cot 'x
y = tan ' x
si (- ₴.) si (0, m.)

- 11

y = cot x

Restricted domain
y= x of cot x is (0, m.)

55/60
Trigonome tric Functions a n d Their Inverses

y= csex

see x y= x

Range of c s c r
sec x Range of es x'
i sT
0. a】y *sh.
-1

k/cr
y = sec.x Csc I x -1 Restricted domain of
Restricted domain of
sec.ris (0, #).x # = 2

y = csc x

56/60
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Trigonometric Functions and Their Inverses

V 1 + x2
x
tan 0 = x →=
• = tan' x
[
1

57/60
Trigonome tric Functions and Their Inverses
Domain: 1- Sx s 1 Domain: 1-5 x≤ 1 Domain: - 00 < x < oc
Range: - 2S y s " Range: Osys n Range:

→
x

Domain: x 5 - 1or x ₴ 1| Domain: x5 1- or x 2 l Domain: - 0 0 < x < 00

Range: 0≤y≤ y * Range: SyS; 9÷0 Range: aO
C
<

y= arcsec x
Ty = acrese * y = arccotx

FIGURE Graphs of the six basic inverse trigonometric functions. 58/60

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H o m e w o r k 1:

Sample Test Problems (Page 52-53)

24, 31, 41, 44, 45(c)

46, 48, 49, 50, 51, 54