precalculus functions and graphs, mark dugopolski, 4th edition

10
xii Function Gallery FUNCTION gallery... Some Basic Functions and Their Properties Constant Function Identity Function Linear Function Square Function Square-Root Function Cube Function Cube-Root Function Greatest Integer Function y x 1 2 3 –1 –2 –3 1 3 2 5 f (x) = 4 Domain 1 - , 2 Range 546 Constant on 1 - , 2 Symmetric about y-axis Absolute Value Function y x 1 2 3 –1 –2 –3 1 3 2 –2 –1 –3 f (x) = x Domain 1 - , 2 Range 1 - , 2 Increasing on 1 - , 2 Symmetric about origin y x 1 2 3 –1 –2 –3 3 4 5 2 –1 f (x) = 3x + 2 Domain 1 - , 2 Range 1 - , 2 Increasing on 1 - , 2 y x 1 2 3 –1 –2 –3 1 2 3 f (x) = xDomain 1 - , 2 Range 30, 2 Increasing on 30, 2 Decreasing on 1 - , 04 Symmetric about y-axis y x 1 2 3 –1 –2 –3 1 2 3 4 5 f (x) = x 2 Domain 1 - , 2 Range 3 - 0, 2 Increasing on 30, 2 Decreasing on 1 - , 04 Symmetric about y-axis y x 1 2 3 4 5 1 2 3 f (x) = x Domain 30, 2 Range 30, 2 Increasing on 30, 2 y x 1 2 –2 2 6 8 4 –4 –2 –6 –8 f (x) = x 3 Domain 1 - , 2 Range 1 - , 2 Increasing on 1 - , 2 Symmetric about origin y x 1 2 3 –1 –2 –3 1 2 –2 –1 f (x) = x 3 Domain 1 - , 2 Range 1 - , 2 Increasing on 1 - , 2 Symmetric about origin y x –1 –3 –2 –3 1 2 3 –2 2 1 f (x) = [[ x]] Domain 1 - , 2 Range 5n ƒ n is an integer 6 Constant on 3n, n + 12 for every integer n

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Some Basic Functions and Their PropertiesSome Inverse FunctionsPolynomial FunctionsSome Basic Rational FunctionsSome Basic Functions of Algebra with TransformationsThe Sine and Cosine FunctionsTrigonometric FunctionsInverse Trigonometric FunctionsFunctions in Polar Coordinates

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Page 1: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

xii Function Gallery

FUNCTION

gallery... Some Basic Functions and Their Properties

Constant Function Identity Function Linear Function

Square Function Square-Root Function

Cube Function Cube-Root Function Greatest Integer Function

y

x1 2 3–1–2–3

1

3

2

5 f(x) = 4

Domain 1-�, �2Range 546Constant on 1-�, �2Symmetric about y-axis

Absolute Value Function

y

x1 2 3–1–2–3

1

3

2

–2

–1

–3

f(x) = x

Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin

y

x1 2 3–1–2–3

3

4

5

2

–1

f(x) = 3x + 2

Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2

y

x1 2 3–1–2–3

1

2

3

f(x) = ⏐x⏐

Domain 1-�, �2Range 30, �2Increasing on 30, �2Decreasing on 1-�, 04Symmetric about y-axis

y

x1 2 3–1–2–3

1

2

3

4

5

f(x) = x2

Domain 1-�, �2Range 3-0, �2Increasing on 30, �2Decreasing on 1-�, 04Symmetric about y-axis

y

x1 2 3 4 5

1

2

3 √f(x) = x

Domain 30, �2Range 30, �2Increasing on 30, �2

y

x1 2–2

2

6

8

4

–4

–2

–6

–8

f(x) = x3

Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin

y

x1 2 3–1–2–3

1

2

–2

–1

√f(x) = x3

Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin

y

x–1–3

–2

–3

1 2 3–2

2

1f (x) = [[ x]]

Domain 1-�, �2Range 5n ƒ n is an integer6Constant on 3n, n + 12 for every integer n

Page 2: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

Function Gallery xiii

FUNCTION

gallery... Some Inverse Functions

Linear

Powers and Roots

y

x–1–2–3–4–1

–2

–3

–4

1 2 3

2

1

3

f –1(x) = x – 3

f (x) = x + 3

f (x) = 5x

f –1(x) = x5

y

x–1–4–1

–4

1 2 3

2

1

3

y

x–1–2–3–4–1

–2

–3

1 2 3

2

1

3

f –1(x) = x+32

f (x) = 2x – 3

x

y

–1

–1

1 2 3 4

2

1

3

4

for x ≥ 0 f (x) = x2

f –1(x) = x

f (x) = x3

–1–2–3–4 1 2 3 4

y

–1

–2

–3

–4

2

1

3

4

f –1(x) = x 3

x

for x ≥ 0

f –1(x) = x

f (x) = x4

4

x

y

–1

–1

1 2 3 4

2

1

3

4

Page 3: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

xiv Function Gallery

FUNCTION

gallery... Polynomial Functions

Linear: f1x2 = mx + b

Quadratic: f1x2 = ax 2 + bx + c or f1x2 = a1x - h22 + k

y

x1–1

1

3

2

–1

–2

–3

f(x) = x3y

x1 2–2

1

3

2

–1

–2

–3

f(x) = x3 – xy

x1–1

1

3

2

–2

–3

f(x) = –x3 + 4x

Quartic or Fourth-Degree: f1x2 = ax 4 + bx 3 + cx 2 + dx + e

Cubic: f1x2 = ax 3 + bx 2 + cx + d

y

x1 2 3–1–2–3

1

3

2

–2

–1

–3

f(x) = x

Slope 1, y-intercept 10, 02

y

x1 2 3–1–2–3

1

3

2

–2

–1

f(x) = 3x – 2

Slope 3, y-intercept 10, -22

y

x1 2 3 4–1–2

1

3

4

2

–1

f(x) = –2x + 4

Slope -2, y-intercept 10, 42

y

x1 2 3–1–2–3

1

3

4

2

–1

f(x) = x2

Vertex 10, 02Range 30, �2

y

x1 2 4–2

1

3

2

–3

–4

f(x) = (x – 1)2 – 4

Vertex 11, -42Range 3-4, �2

y

x2–1–2–4

1

3

4

2

–1

–2

f(x) = –x2 – 2x + 3

Vertex 1-1, 42Range 1-�, 44

y

x1 2–1–2

1

3

4

5

2

f(x) = x4 y

x1–1

2

1

f(x) = x4 – x2 y

x1 2–1–2

1

3

–1

–2

f(x) = –x4 + x2 + 2

Page 4: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

Function Gallery xv

FUNCTION

gallery... Some Basic Rational Functions

Horizontal Asymptote x-axis and Vertical Asymptote y-axis

Various Asymptotes

y

x1 2 3–1–3

1

2

3

–1

f(x) = x–1

y

x1–1–2–3

1

–2

–3

–1

f(x) = – x–1

y

x21 3–3 –2 –1

1

2

f(x) =x2––1

y

x1 2 3–1–2–3

1

3

–2

–1

f(x) = x——–2x – 1

y = 2

y

x2 3–3

1

2

3

–2

–3

–1

f(x) =x2 – 1——–x

x = –1

x = 1

y

x21 3–3 –2 –1

1

–2

–3

–1

f(x) = x2 – 1——–x

y = x

Page 5: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

xvi Function Gallery

FUNCTION

gallery... Exponential and Logarithmic Functions

Exponential: f1x2 = a x, domain (-H , H ), range 10, H2

Logarithmic: f -11x2 = loga1x2, domain 10, H2, range 1-H , H2

y

x1 2 3–1–2

1

3

4

5

6

2

g(x) = 2x

y

x1 2–1–2

1

3

4

5

6

2

h(x) =x1–

2⎞⎠

⎞⎠

y

x1 2–1–2

1

3

4

5

6

2

j(x) = ex

y

x1 2 3 4 5 6 7

1

–2

–1

2

g–1(x) = log2(x) y

x1 2 3 4 5 6 7

1

–2

–1

2h–1(x) = log1/2(x)

y

x1 2 3 4 5 6 7

1

–2

–1

2 j –1(x) = ln(x)

Increasing on 10, �2 x-intercept 11, 02Decreasing on 10, �2

x-intercept 11, 02Increasing on 10, �2 x-intercept 11, 02

Increasing on 1-�, �2y-intercept 10, 12 Decreasing on 1-�, �2

y-intercept 10, 12 Increasing on 1-�, �2y-intercept 10, 12

Page 6: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

Function Gallery xvii

y

x–1–1

–2

–3

1 2 3–2

2

1

3

Quadratic

4

y = (x – 3)2

y = –x2

y = x2

y

x–1

–2

–3

1 2 3

2

1

3

Cubic

4 5

y = –x3

y = x3

y = (x – 4)3

y

x–1–3–1

–2

–3

1 2 3

2

1

3

Absolute value

y = ⏐x⏐

y = ⏐x⏐ – 3

y = ⏐x – 2⏐

Exponential

y

x

–11

1

2

2

–2

y = 2x

y = 2 – x

y = –2x

y

x–1–1

–2

1 2

2

1

3

Logarithmic

–2

4

y = log2(x)

y = log2(x) + 3

y = log2(x + 4)

y

x–1–3–1

–2

–3

1 2 3–2

1

3

Square root

y = x + 4

y = x + 3 – 4

y = x

Reciprocal

y

x–1–1

1

1

y = – 1–x

y = 1–x

Rational

y

x–1–1

1

1

2

–2y = – 1—

x2

y = 1—x2

y

x–1

–2

–3

1 2 3

2

1

3

Fourth degree

4 5

y = x4

y = (x – 4)4

y = – x4 – 1

y

x–1–1

1

1

Semicircle

y = – 4 – x2

y = 4 – x2√

y

x–1

–2

–3

1 2 3

2

1

3

Greatest integer

4 5

y = [[x]]

y = [[x – 3]]

FUNCTION

gallery... Some Basic Functions of Algebra with Transformations

y

x–1–3–1

–2

–3

1 2 3–2

2

1

3

Linear

y = x

y = x + 3

y = – x

Page 7: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

xviii Function Gallery

FUNCTION

gallery... The Sine and Cosine Functions

y

x

–1

1

y = sin x

π–2

π 3π—2

20 π

Amplitude 1

πPeriod 2Phase shift 0

y

D – A

D

D + A

x0 C

2π—B

2π—B

C +

Amplitude A

PeriodPhaseshift C

y = A sin[B(x – C)] + D

A, B, C, D > 0

y

x

–1

1

y = cos x

π–2

π 3π—2

20 π

Amplitude 1

πPeriod 2Phase shift 0

y

D – A

D

D + A

x0 C

2π—B

2π—B

C +

Amplitude A

PeriodPhaseshift C

y = A cos[B(x – C)] + D

A, B, C, D > 0

FUNCTION

gallery... Periods of Sine, Cosine, and Tangent 1 B 7 1 2 y

x

1

–1

Period2�B

y = sin x

y = sin (Bx)

2�� 2�B

�B

Fundamental cycles

y

x

1

–1

Period2�B

y = cos x

y = cos (Bx)

2�

� 2�

B�B

y

x

1

–1

y = tan (Bx)Period

�B

y = tan x

�2

�2

�2B

�2B

––

Page 8: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

Function Gallery xix

FUNCTION

gallery... Trigonometric Functions

Domain(k any integer)RangePeriod

Fundamental cycle

1- q , q2 3-1, 14

2p

30, 2p4

1- q , q2 3-1, 14 2p

30, 2p4

x �p

2+ kp

1- q , q2 p

c-p2

, p

2d

Domain (k any integer)Range Period Fundamental cycle

x � kp

1- q , -14 � 31, q22p30, 2p4

x �p

2+ kp

1- q , -14 � 31, q2 2p 30, 2p4

x � kp 1- q , q2 p

30, p4

y

x

–4

4321

–1–2–3

π 2π

y = csc(x)

y

x

–1

1

π 2π

y = sin(x)

y

x

–4

4321

–1–2–3

π 2π

y = sec(x)

y

x

–1

1

π 2π

y = cos(x)

y

x

–4

4321

–1–2–3

ππ–2

y = cot(x)

y

x

–4

4321

–1–2–3

π–2

π–2

–y = tan(x)

Page 9: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

xx Function Gallery

FUNCTION

gallery... Inverse Trigonometric Functions

1–1

y

x

Domain [–1, 1]

Range –2– ⎡

⎣, –2⎡⎣

� �

y = sin–1 x

�2

�2

Domain [–1, 1]

Range [0, p]

y

x–1 1

y = cos–1 x�

2 1–1

y

x

Domain

Range –2– , –2

� �

y = tan–1 x�2

�2

(–�, �)⎞⎠⎝

Domain Range

y

x

–⎡⎣

⎡⎣, 0 0,)) �

–1 1

–2�–2

y = csc–1 x�2

�2

(–�, –1] � [1, �)

1–1

y

x

Domain (– , –1] [1, )

Range ⎡⎣

⎡⎣0, –2

, –2

�) )��

���

2

y = sec–1 x

1–1

y

x

Domain (– , )

Range (0, )� �

y = cot–1 x�2

Page 10: Precalculus Functions and Graphs, Mark Dugopolski, 4th Edition

Function Gallery xxi

FUNCTION

gallery... Functions in Polar Coordinates

–10 –8 –6 –4 –2 2 4 6 8–2

2

4

6

8

–4

–6

–8

–10

r = θθ ≥ 0

Spiral

10

10

a–2

y

xa–2

a

a–2

r = a cos θa > 0

Circle

a–2

y

xa–2

a–2

a

r = asin θa > 0

Circle

a

y

xa–a

–a

r = a cos 3θa > 0

Three-Leaf Rose

a

y

x–a

–a

–2a

Cardioid

a > 0r = a(1 – cos θ)

a

y

xa–a

–a

r = a cos 2θa > 0

Four-Leaf Rose

a

y

x–a

–a

–(a + b)

Limaçon

b > a > 0r = a – b cos θ

y

x√⎯a– √⎯a

r2 = a cos 2θa > 0

Lemniscate

a

y

x–a

–a

–(a + b) a – b

Limaçon

a > b > 0r = a – b cos θ