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
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
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
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
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
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
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
––
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)
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
�
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
