Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
QUAN 111 - Mathematics for Economics and Finance
Week 3: Functions of One Variable
Yiğit Sağlam
School of Economics and Finance
Victoria University of Wellington
July 2022
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Summation Notation
Definition
The capital Greek letter sigma Σ denotes the summation symbol:
6
X
xi = x1 + x2 + x3 + x4 + x5 + x6
i=1
where
▶ i is the index of summation,
▶ 1 (in this example) is the lower limit,
▶ 6 (in this example) is the upper limit.
Let’s see some examples...
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Summaton Notation
Example
10
X
i=1
i = 1 + 2 + . . . + 10 =
10(10 + 1)
= 55
2
Example
2
X
√
√
√
√
√
√
2 k + 2 = 2 −2 + 2 + 2 −1 + 2 + 2 0 + 2 + 2 1 + 2 + 2 2 + 2
k=−2
√
√
√
√
√
√
√ =2 0+2 1+2 2+2 3+2 4= 2 3+ 2+ 3
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Rules for Sums
Definition
Two very useful rules when manipulating sums are:
▶ Additivity Property:
N
X
(ai + bi ) =
i=1
N
X
ai +
i=1
N
X
bi
i=1
▶ Homogeneity Property:
N
X
i=1
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QUAN111 - Week 3
c ai = c
N
X
ai
i=1
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Rules for Sums
Example
Take a look at the definition of mean:
µx =
N
1 X
1
xi =
(x1 + x2 + . . . + xN )
N i=1
N
Example
Take a look at the definition of variance:
σx2 =
Yiğit Sağlam
QUAN111 - Week 3
N
1 X
(xi − µx )2
N i=1
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Useful Formulas
Definition
The sum of successive numbers is:
N
X
i = 1 + 2 + 3 + ... + N =
i=1
N (N + 1)
2
Example
20
X
i=1
Yiğit Sağlam
QUAN111 - Week 3
i = 1 + 2 + 3 + . . . + 20 =
20(20 + 1)
= 210
2
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Useful Formulas
Example (Bond Value)
Calculate the value of a corporate bond with annual interest rate of 7%,
making annual interest payments for 3 years, after which the bond matures and
the principal must be repaid $1000. Assume a discount rate of 5%.
Present Value of Face Value =1000
1
= 863.84
(1 + 0.05)3
Annual Payments =1000 (0.07) = 70(every year)
Present Value of Annual Payments =
3
X
70
= 190.63
t
1.05
t=1
Bond Value (today) =863.84 + 190.63 = 1054.47
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Useful Formulas
Example (Perpetuity)
A perpetuity is an annuity in which we make regular payments forever. One
example is fixed coupon payments on permanently invested bonds. Calculate
the value of a perpetuity making annual interest payments $70 forever,
assuming a discount rate of 5%.
∞
X
70
70
70
70
=
+
+
+ ...
t
2
3
1.05
1.05
1.05
1.05
t=1
70
70
=
=
= 1400
1.05 − 1
0.05
Value of Perpetuity =
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Summation Notation
Useful Formulas
Problem
Solve the following problems:
▶ Calculate the value of a perpetuity making annual interest payments $25
forever, assuming a discount rate of 2%.
10 P
1
1
▶
− i+1
i
i=1
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Logic Statements
A Few Aspects of Logic
Definition
▶ Assertions that are either true or false are called statements or
propositions.
▶ The implication arrow points in the direction of the logical implication
P ⇒ Q.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Logic Statements
A Few Aspects of Logic
Example
True of False?
▶ x > −1 ⇒ x > 0
▶ x2 + y 2 = 0 ⇒ x = 0 and y = 0
▶ x = 0 and y = 0 ⇒ x2 + y 2 = 0
▶ x > y2 ⇒ x > 0
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Logic Statements
A Few Aspects of Logic
Definition
▶ P is a sufficient condition for Q means P ⇒ Q.
▶ Q is a necessary condition for P means P ⇒ Q.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Logic Statements
A Few Aspects of Logic
Example
Necessary or sufficient condition for 2x + 5 ≥ 13?
▶ x≥0
▶ x ≥ 50
▶ x≥4
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Set Theory
Set Theory
Definition
▶ A collection of objects is called a set.
▶ These objects are referred to as elements.
Example
Roll a die. The collection of outcomes is the set A = {1, 2, 3, 4, 5, 6}.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Set Theory
Set Theory
Definition
Consider two set A and B:
▶ A = B if the two sets have the same elements.
▶ If x is an element of set A,then we use x ∈ A.
▶ If set B has all the elements of set A, then A is a subset of B (A ⊆ B).
Example
A = {2, 3, 4} , B = {2, 5, 6} , C = {5, 6, 2} , D = {6}.
▶ Is it true that B = C?
▶ Is it true that D ⊆ C?
▶ Is it true that A ⊆ C?
▶ Is it true that B ⊆ C?
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Set Theory
Set Theory
Definition
Consider two set A and B:
▶ The union of sets A and B is:
A∪B = x
x ∈ A or x ∈ B
▶ The intersection of sets A and B is:
A ∩ B = x x ∈ A and x ∈ B
▶ The difference of set A from B is (A minus B):
A\B = x x ∈ A but x ∈
/B
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Set Theory
Set Theory
Example
Suppose that we roll a die and we collect outcomes in the following sets:
A = {1, 2, 3, 4} , B = {5, 6} , C = {1, 3, 5} , D = {2, 4, 6}.
▶ S = {1, 2, 3, 4, 5, 6} is called the sample space, as it covers all possible
outcomes.
▶ A and B are disjoint sets, because A ∩ B = ∅.
▶ Are C and D disjoint? What about A and C?
▶ B = Ac is a complement of A because S\A = B.
▶ Is C a complement of D? What about D = B c ?
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Set Theory
Venn Diagram
A
(A ∩ B)\C
(A ∩ C)\B
A∩B∩C
B
C
(B ∩ C)\A
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Graphing Functions
Definition of Function
Example
▶ A (real-valued) function of a real variable x with domain D is a rule that
assigns a unique real number to each real number x in D.
▶ As x varies over the whole domain, the set of all possible resulting values
f (x) is called the range of f .
f : D→R
f (x) = y, x ∈ D and y ∈ R.
▶ x ∈ D is called the independent variable or the argument.
▶ y ∈ R is called the dependent variable.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Graphing Functions
Function of One Variable
Example
Suppose f (x) = 10, for all x ∈ R. Calculate:
▶ f (0) = 10 .
▶ f (−3) = 10 .
▶ f (a + h) − f (a) = 10 − 10 = 0 .
Example
Suppose f (x) = x2 + 1, for all x ∈ R. Calculate:
▶ f (0) = 02 + 1 = 1 , f (−1) = (−1)2 + 1 = 2 .
▶ f (x) = −f (x)? What about f (2x) = 2f (x)?
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Graphing Functions
Coordinate System
Definition
▶ The xy-plane is the rectangular (or Cartesian) coordinate system
obtained by drawing two perpendicular lines, called coordinate axes.
▶ x-axis is the horizontal axis and y-axis is the vertical axis.
▶ The intersection point O is called the origin.
Figure 1: Coordinate System
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QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Graphing Functions
Coordinates
Definition
▶ (3, 4) is called the coordinates of P .
▶ (3, 4) is an ordered pair: we cannot switch these two numbers.
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QUAN111 - Week 3
Figure 2: Coordinate System with points
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Graphing Functions
Graph of a Function
Definition
▶ The graph of a function f is simply the set of all points (x, f (x)), where x
belongs to the domain of f .
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Linear Functions
Definition
Linear functions are of the following form:
y = ax + b
where a, b are parameters. The slope is a, and the intercept is b.
Figure 3: Linear Function
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Slope of a Straight Line
Definition
The slope of a straight line is:
a=
y2 − y1
, x1 ̸= x2
x2 − x1
where (x1 , y1 ) and (x2 , y2 ) are two distinct points on the line.
Figure 4: Slope of a Linear Function
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Slope of a Straight Line
Definition
▶ Point-Slope Formula of Straight Line: The equation of the straight line
passing through (x1 , y1 ) with slope a is:
y − y1 = a (x − x1 )
▶ Point-Point Formula of Straight Line: Let (x1 , y1 ) and (x2 , y2 ) are two
distinct points where the straight line passes through.
1. Compute the slope:
y2 − y1
, x1 ̸= x2
x2 − x1
2. The formula for the straight line is then:
y2 − y1
y − y1 =
(x − x1 )
x2 − x1
a=
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Slope of a Straight Line
Example
Find the straight line equation that passes through (2, 3) and (5, 8).
1. Compute the slope:
a=
y2 − y1
8−3
=
= 5/3
x2 − x1
5−2
2. The formula for the straight line is then:
y2 − y1
y − y1 =
(x − x1 )
x2 − x1
5
y − 3 = (x − 2)
3
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Linear Inequalities
Example
Sketch the xy-plane for 2x + y ≤ 4
Figure 5: Linear Inequality
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Linear Functions
Linear Inequalities
Example
Sketch the xy-plane for the budget constraint p x + q y ≤ m, where x ≥ 0, and
y ≥ 0.
Figure 6: Linear Inequality
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Quadratic Functions
Quadratic Functions
Definition
Quadratic functions, also known as a parabola, are of the following form:
f (x) = a x2 + b x + c, a, b, c are constants, a ̸= 0.
The maximum/minimum point P is called the vertex of the parabola.
Figure 7: Quadratic Functions
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Quadratic Functions
Quadratic Functions
Definition
We can always rewrite a quadratic function as follows:
2
b2 − 4ac
b
−
a x2 + b x + c = a x +
2a
4a
b
▶ If a > 0, then the quadratic function has its minimum at x = − 2a
.
b
▶ If a < 0, then the quadratic function has its maximum at x = − 2a
.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Quadratic Functions
Quadratic Functions
Example
Sketch the graph of f (x) = x2 − 4x + 3
Figure 8: Quadratic Functions
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Quadratic Functions
GoSoapBox Problem
Event Code:
vuwquan111
Enter this code at: https://app.gosoapbox.com
Problem
Consider the following function in variable x:
f (x) = x2 + 2x + 4
What is lowest value for y = f (x)?
a) -1
b) 1
c) 3
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Quadratic Functions
Polynomial Functions
Definition
Polynomial functions, which contain all forms of linear, quadratic, and cubic
functions, are of the following general form:
P (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 , a’s are constants, an ̸= 0.
This form represents a polynomial of degree n with coefficients
{an , an−1 , an−2 , . . . , a1 , a0 }.
▶ A polynomial of degree n has at most n real solutions, called the roots.
▶ Fundamental Theorem of Algebra: Every polynomial can be written as a
product of polynomials of degree 1 or 2.
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Other Functions
Power Functions
Definition
Power functions have the following form:
f (x) = Axr , x ≥ 0, A and r are constants.
Figure 9: Power Functions
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QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Other Functions
Exponential Functions
Definition
Exponential functions have the following form:
f (x) = Aax , a ≥ 0, A is constant.
A special case is the natural exponential function:
f (x) = ex , e = 2.718281828459045 . . .
Figure 10: Exponential Functions
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Other Functions
Logarithmic Functions
Definition
If eu = x, then u is the natural logarithm of x and is denoted by:
u = ln x
More formally, the logarithmic function has the following form:
g(x) = ln x, x > 0
Figure 11: Logarithmic Functions
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Other Functions
Rules of the Natural Logarithmic Functions
Definition
▶ ln(xy) = ln x + ln y,for x, y > 0
▶ ln xy = ln x − ln y, for x, y > 0
▶ ln (xp ) = p ln x, for x > 0
▶ ln 1 = 0, ln e = 1, x = eln x ,
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QUAN111 - Week 3
ln (ex ) = x
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Other Functions
Logarithmic Functions with Bases Other than e
Definition
If au = x, then u is the natural logarithm of x to base a and is denoted by:
u = loga x
▶ loga (xy) = loga x + loga y, for x, y > 0
▶ loga xy = loga x − loga y, for x, y > 0
▶ loga (xp ) = p loga x, for x > 0
▶ loga 1 = 0, loga a = 1
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Inverse Functions
Inverse Functions
Definition
Let the function f have a domain A and range B, which consists of all the
points f (x) can take for x ∈ A. The function f is said to be one-to-one in A
if f never has the same value at any two different points in A.
Definition
Let f be a function with domain A and range B:
f : A→B
If and only if f is one-to-one, it has an inverse function g with domain B and
range A:
g(y) = x ⇔ y = f (x), (x ∈ A, y ∈ B)
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Inverse Functions
Inverse Functions
Example
The inverse of y = f (x) = ex is the natural logarithmic function:
x = g(y) = ln y
Yiğit Sağlam
QUAN111 - Week 3
VUW
Miscellaneous
Set theory
Functions of One Variable
Functional Forms
Inverse of a Function
Inverse Functions
Inverse Functions
Example
What is the inverse function g(y) of the following function y = f (x)?
y = f (x) = 2x5 + 1, x ≥ 0
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QUAN111 - Week 3
VUW