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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Functions in terms of Set Notation
MATH0201
BASIC CALCULUS
A Library of Basic Functions
Polynomial Functions
Functions
Rational Functions
Piecewise–Defined Function
Dr. WONG Chi Wing
Department of Mathematics, HKU
Absolute Value Function
Trigonometric Function
Invertible Functions
Reference § 2.1–4 of Barnett.
MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
However, not every equation in two variables define a function.
To summarize, a function can be represented by
I
Tables
I
Graphs
I
Formulas, or more general, an equation of two variables.
Example 2
The equation x 2 + y 2 = 2 does not
define a function.
Given that x = 1, there corresponds
the real values y = ±1. a
Example 1 (Functions Defined by
Equation)
a
The equation 3x + 2y = −1 defines a
function.
Every x is mapped to the number
y = −(3x + 1)/2. a
a
Read Example 2(A) (p.47)
Read Example 2(B) (p.47)
Graph of x 2 + y 2 = 2
Theorem 3 (Vertical Line Test for Function
(p.48))
An equation specifies a function if each vertical
line in the coordinate system passes through, at
most, one point on the graph of the equation;
otherwise the equation does not specify a function.
Graph of 3x + 2y = −1
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Functions in terms of Set Notation
Functions in terms of Set Notation
Example 5
Let f : N → R be defined by f (x) = x 2 .
Definition 4 (Function (p.46))
Domain : N
Codomain : R
Range
: {1, 22 , 32 , 42 , . . .} = {n2 : n ∈ N}
A function is a correspondence between two sets of elements
such that to EACH element in the first set X , there corresponds
ONE and ONLY ONE element in the second set Y .
If we denote the function by f , then we may write f : X → Y .
We call X and Y the domain and the codomain of the function
respectively; and the set of the corresponding elements in Y is
called the range of the function.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Functions in terms of Set Notation
Functions in terms of Set Notation
Example 6
If the graph of a function is given, then its
range is the projection of the graph on the
y –axis.
Let f : R → R be defined by f (x) = x 2 .
Domain : R
Codomain : R
Range
: [0, ∞)
Example 7
The range of function below is the blue segment.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Functions in terms of Set Notation
Functions in terms of Set Notation
If the domain of the function f (x) is not specified,
then we shall assume the natural domain, i.e., the
set of all real–number replacements of the independent variable that produce real values of the dependent variable. (p.48)
Definition 9 (Increasing/Decreasing Functions (p.267))
We say that a function f is increasing on an interval (a, b) if
f (x1 ) < f (x2 ) whenever a < x1 < x2 < b,
Example 8
and f is decreasing on an interval (a, b) if
Find the natural domain of 1
f (x1 ) > f (x2 ) whenever a < x1 < x2 < b,
1. f (x) = x 2 + x − 1,
√
2. g(x) = x − 1,
Further, we say that f is monotone on an interval (a, b) if it is
either increasing or decreasing there.
3. h(x) = 1/(x − 2).
1
Read Example 3 (p.48), 5 (p.50).
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Polynomial Functions
Example 10 (Cost and Demand Function (p.51))
Linear Function
The cost C of manufacturing x units of a product may be given by
A linear function takes the form
f (x) = ax + b,
C(x) = a + bx,
where a, b > 0;
while the price of a product p when x units of that product are available
may be given by
where a and b are fixed reals.
p(x) = m − nx
Domain : R
Codomain : R
Range
:R
Blank
f (x) is increasing if a > 0
f (x) is decreasing if a < 0
where m, n > 0.
Example 11 (Fahrenheit–Celsius Temperature Conversion)
The Fahrenbeit temperature scale (y ◦ F ) is related to the Celsius
temperature scale (x ◦ C) by
9
y = 32 + x.
5
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Polynomial Functions
Quadratic Function
Example 12 (Velocity of a Free Falling Object)
The velocity of a free falling object is given by
A quadratic function takes the form
v (t) = v (0) + gt
f (x) = ax 2 + bx + c,
where a, b, and c are fixed reals.
where g is the acceleration due to gravity (in ms−2 ).
Domain
Codomain
Range
Example 13 (Zero Order Reaction)
In a zero order reaction, the concentration of a reactant A is
given by
[A](t) = [A](0) − kt
:
:
:
R
R
[k , ∞) if a > 0
(−∞, k ] if a < 0
Blank
Neither increasing nor decreasing
where k is the reaction rate.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Polynomial Functions
Example 14 (Revenue and Profit Function (p.51))
Example 15 (Distance Travel of a Free Falling Object)
If the cost function and the demand function of a product are
respectively given by
The height of the Leaning Tower of Pisa is 55m. When Galilei
dropped a cannon ball from the top, the height of the cannon
ball after t seconds was
C(x) = a + bx and p(x) = m − nx.
h(t) = 55 −
The revenue function is
R(x) = xp(x) = mx − nx 2 .
g 2
t .
2
Example 16 (Energy stored in a spring)
The profit function is
If a spring with natural length `0 is compressed/stretched to the
length `, then the energy stored E is
P(x) = R(x) − C(x) = −nx 2 + (m − b)x − a.
E(`) =
Read Example 7 (p.52).
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k
(` − `0 )2 .
2
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Polynomial Functions
Cube Function
Third–degree Polynomial Function
The cube function is defined by the formula
A third–degree polynomial function takes the form
f (x) = x 3 .
f (x) = ax 3 +bx 2 +cx +d,
Domain : R
Codomain : R
Range
:R
Blank
f (x) = x 3 is increasing
where a, b, c, and d are fixed reals.
Domain : R
Codomain : R
Range
:R
They are neither increasing nor decreasing in general.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Polynomial Functions
Square Root Function
Example 17 (Compound Interest)
The square root function is defined by the formula
√
f (x) = x.
If a principal P is invested at an annual rate r compounded
annually, the the amount A at the end of the n–th year is
P(r ) = A (1 + r )n .
n=1:
n=2:
n=3:
n=4:
Domain : [0, ∞)
Codomain : R
Range
: [0, ∞)
Blank √
f (x) = x is increasing
P(r ) = A(1 + r )
P(r ) = A 1 + 2r + r 2
P(r ) = A 1 + 3r + 3r 2 + r 3
P(r ) = A 1 + 4r + 6r 2 + 4r 3 + r 4
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Polynomial Functions
Rational Functions
Cube Root Function
Definition 18 (Rational Function (p.88))
A rational function is any function that can be written in the form
The cube root function is defined by the formula
√
f (x) = 3 x.
f (x) =
n(x)
,
d(x)
d(x) 6= 0
where n(x) and d(x) are polynomials.
Domain of f (x) is the set of all the reals x such that d(x) 6= 0.
Domain : R
Codomain : R
Range
:R
Blank √
f (x) = 3 x is increasing
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Rational Functions
Piecewise–Defined Function
Definition 20 (Piecewise–defined Function (p.64))
Example 19 (Lennard–Jones Potential)
A piecewise–defined function is defined by rules for different parts of the domain.
The potential energy V when two molecules approach each
other from a long distance R apart is modeled by
Example 21 (Example 6 (p.65))
Easton Utilities uses the rates shown in the table below to compute the monthly cost of
natural gas for each customer.
σ 12 − σ 6 R 6
.
V (R) = 4ε ·
R 12
Note that
I
V (σ) = 0.
I
The minimum value of
V is −ε, and it’s
achieved when
R = 21/6 σ.
First 5 CCF : $ .7866 CCF-1
Next 35 CCF : $ .4601 CCF-1
Over 40 CCF : $ .2508 CCF-1
The cost of consuming x CCF is

 .7866x
3.933 + .4601(x − 5)
C(x) =

20.0365 + .2508(x − 40)
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if 0 ≤ x ≤ 5,
if 5 < x ≤ 40,
if x > 40.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Absolute Value Function
Absolute Value Function
Definition 22
The absolute value of a real number a, denoted by |a|, is
defined by
a
if a ≥ 0,
|a| =
−a
if a < 0.
Theorem 24
For any real numbers a and b,
1. |a| ≥ 0
Example 23
2. |a| = | − a|
3. |ab| = |a||b|
1
4. a1 = |a|
if a 6= 0
√
5. a2 = |a|.
(a) |3| = 3.
(b) | − 2| = 2.
(c) ||3 − 11| − |3|| = 5.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Absolute Value Function
Absolute Value Function
Example 25
Example 26
The graph of y = |2x − 1| is
The graph of y = |x 2 − 3x + 2| is
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Absolute Value Function
Absolute Value Function
Theorem 27
Theorem 30
For any a ≥ 0, solutions of |x| = a are a and −a.
Let d > 0. Then the solutions of
1. |x| < d are exactly the real numbers satisfying −d < x < d,
2. |x| ≤ d are exactly the real numbers satisfying −d ≤ x ≤ d,
3. |x| > d are exactly the real numbers satisfying x > d or
x < −d,
4. |x| ≥ d are exactly the real numbers satisfying x ≥ d or
x ≤ −d.
Example 28
The solution set of
1. |x| = 11 is {11, −11}.
2. |x| = 0 is {0}.
3. |x| = −1 is ∅.
Example 29
Solve |3x − 2| = 1.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Absolute Value Function
Trigonometric Function
Angle
An angle is formed by rotating a ray about its endpoint.
Example 31
Solve the following inequalities
(a)
|2x − 1| < 3
(b)
|3x + 2| ≥ 4.
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I
The endpoint of the ray is called the vertex.
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The initial position of the ray is called the initial side.
I
The final position of the ray is called the terminal side.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
A Library of Basic Functions
Trigonometric Function
Trigonometric Function
Sine and Cosine of an Angle
Definition 32
In the Cartesian plane, an angle is said to be in standard
position if its vertex is at the origin and its initial side is along
the positive x–axis.
Consider the unit circle in a coordinate system centered at the
origin.
I Terminal side of an
angle θ in standard
position will pass
through the circle at a
point P.
Angles can be represented by (real)
numbers:
I
I
A counterclockwise rotation has a
positive measure while a
clockwise rotation has a negative
measure.
If there is more than one
complete rotation, then the
magnitude of an angle can be
greater than 360 degrees.
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Abscissa of P is called
the cosine of θ, denoted
by cos θ.
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Ordinate of P is called
the sine of θ, denoted
by sin θ.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
A Library of Basic Functions
Invertible Functions
Trigonometric Function
Definition 34 (One–to–one Function and Inverse Function
(cf p.106–107))
Definition 33 (Sine, Cosine, and Tangent Functions)
A function is said to be one–to–one if each range value corresponds
to exactly one domain value.
The function which sends every real number x to the sine of an
angle with magnitude x ◦ is called the sine function, denoted by
sin x.
If f is a one–to–one function, then the inverse of f , denoted by f −1 , is
the function formed by interchanging the independent and dependent
variables for f . Thus, if (a, b) is a point on the graph of f , then (b, a) is
a point on the graph of f −1 .
Similarly, the function which sends every real number x to the
cosine of an angle with magnitude x ◦ is called the cosine
function, denoted by cos x.
sin x
The tangent function is defined by tan x =
.
cos x
Remark 1
It’s clear that the domain of f is exactly the range of f −1 while the
range of f is exactly the domain of f −1 .
Theorem 35
If f is a monotone function on an interval (a, b), then it has inverse.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Invertible Functions
Invertible Functions
Finding Inverse Graphically
Given that f has an inverse. Then the graph of f −1
is the reflection of the graph of f along the line y =
x.
Example 36
Both the linear function f (x) = ax + b (a 6= 0) and the cube
function g(x) = x 3 have inverses.
f −1 (x) = (x − b)/a
g −1 (x) =
√
3
x
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MATH0201 BASIC CALCULUS
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MATH0201 BASIC CALCULUS
Invertible Functions
Invertible Functions
Example 36
The square function f (x) = x 2 is not one–to–one and hence it
has no inverse. Actually the reflection of its graph along y = x
fails the vertical line test.
Finding Inverse Analytically
Let f be a one–to–one function. If a single x can
be solved for the equation y = f (x) (in terms of y ),
then f has an inverse.
Example 37
Find the inverse of f (x) =
√
3x − 1, if any.
Class Practice 7
Find the inverse of 4x 3 − 7, if any.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Invertible Functions
Invertible Functions
Example 38
While f (x) = x 2 is not one–to–one, its restriction to [0, ∞) is
increasing and hence has an inverse.
Example 39
The sine function f (x) = sin x has no inverse, but its restriction
to [−90, 90] has an inverse.The inverse is denoted by arcsin x.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Invertible Functions
Invertible Functions
Example 40
Example 41
The cosine function f (x) = cos x has no inverse, but its
restriction to [0, 180] has an inverse.The inverse is denoted by
arccos x.
The tangent function f (x) = tan x has no inverse, but its
restriction to (−90, 90) has an inverse.The inverse is denoted
by arctan x.
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