Quantitative methods
Lecture 2
Mgr. Radmila Krkošková, Ph.D.
BAKVM

A real function of one real variable
•A function is a relation f between two sets X and Y such that each x from X is related to exactly
one y from Y.
•We write y = f(x).
•Examples:
•
•The set X is called a domain, the set Y is called a range or a co-domain.
•In economics the domain usually consists of non-negative real numbers.

A graph of a function
•A graph of a function is the collection of all ordered pairs (x,y) displayed in a two-dimensial
plane.
•In a two dimensional plane it as a curve.
•
• Examples:
•Linear function: a line,
•Quadratic function: a parabola,
•Reciprocal function: a hyperbola.
•
•Other functions are represented by more complex curves.
•
•
•
Lecture 2

A graph of a function: examples
•
•
•
•
Lecture 2
graf08

Elementary function properties
•
•
•
•
Lecture 2
• Domain and range,
• Monotonicity (increasing, decreasing, non-increasing, non-decreasing, constant)
• Extremes (local or global maximum or minimum),
• Concavity and convexity,
• Inflection points,
• Bounded vs unbounded function,
• Even functions and odd functions,
• Peridiocity.

Elementary functions
•
•
•
•
Lecture 2
• Linear function: y = ax + b,
• Quadratic function:
• Polynomial function:
• Linear reciprocal function:
• Logarithmic function:
In the logarithmic function, a is the so called a base. The decadic logarithm: a = 10, the natural
logarithm: a = e = 2.718… For a larger than 1, the function is increasing, and for a smaller than 1
it is decreasing.

Graphs of logarithmic functions
•
•
•
•
Lecture 2
graf05 graf37

Elementary functions - continued
•
•
•
•
Lecture 2
• Exponential function:
• Goniometric functions: y = sinx, y = cosx,  y = tgx,
   y = cotgx.
• Cyclometric functions: y = arcsinx, y = arccosx,
   y = arctgx, y = arccotgx.
Notes: exponential and logarithmic functions are inverse to each other. The same applies to
goniometric and cyclometric functions.
•

Graphs of exponential functions
•
•
•
•
Lecture 2
For a >1, the exponential function is increasing, for a < 1 is
decreasing.

Graphs of goniometric functions
Lecture 2
graf33 graf34 graf35 graf36

Polynomials
•
•
•
•
Lecture 2
In economics, various functions, such as demand or supply, are expressed by polynomials. Two main
tasks when dealing with polynomials are transformation of a polyonomial into a product, and to find
roots of a polynomial.
Let            denote a polynomial of a degree n.
Polynomial roots are such values of x that                  .
The equation above can be solved via known formulas or identities such
as                              .

Demand and supply function, equilibrium
•The demand function expresses relationship between a price of a good (P) and a demanded quantity
(Q) by customers. Usually, the demand function is denoted as Q= D(P) or QD , and it is assumed this
function is decreasing.
•The supply function expresses relationship between a price of a good (P) and a supplied quantity
(Q) by sellers. Usually, the supply function is denoted as Q= S(P) or Qs , and it is assumed this
function is increasing.
•A point where demand is equal to supply, ant a market is cleared, is called an equilibrium.
Lecture 2

Demand and supply function, equilibrium – cont.
•
•
•
•
Lecture 2
Graf 31

Solved problem 1
•Find the domain of the function f:                     .
• Solution: the expression under the square root sign must be non-negative, therefore we obtain:
•
•
•
•Hence, the domain is                             .
Lecture 2

Solved problem 2
•Find the domain of the function f:                     .
• Solution: the expression in the logarithm must be positive, therefore we obtain:
•
• We expand the term on the left hand side:
•
•
•From the last inequality it follows that -5 and 5 are the roots that divide the x line into three
intervals. By checking the sign in each interval we obtain the final solution:
•
•
•
•
•
•
Lecture 2

Solved problem 3
•Find the domain of the function f:                       .
• Solution: the expression in the arcsin is bounded by -1 from below and by 1 from above.
Therefore, we obtain:
•
•
• By dividing this inequality into two simple linear inequalities we obtain:
•                   and
•
•Hence, we obtain the solution:
•
Lecture 2

Solved problem 4
•Let us asssume that the demand and the supply functions are given as follows:
,                     .
•Find the equilibrium.
•
•Solution: in the equilibrium both functions are equal:
•
•
•Therefore, we obtain: PE = 6, and QE = 4. Draw both function!
•
•How will the situation change if there is a price floor P = 8?
•
•
•
Lecture 2

Solved problem 5
•Let us asssume that the demand and the supply functions are given as follows:
,                     .
•Find the equilibrium.
•
•Solution: in the equilibrium both functions are equal:
•
•
•
•Therefore, we obtain: PE = 3, and QE = 18. Draw both function!
•
•
Lecture 2

Problems to solve
•1. Find the domain of the following functions:
•
•
•
•
•
Lecture 2

Problems to solve – cont.
•2. Draw a graph of the following functions:
•
•
•
•
•
•
Lecture 2

Problems to solve – cont.
•3. For the given functions of demand and supply find the equilibrium both geometrically and
algebraically:
•
•
•
•
•
•
Lecture 2

•
•
•
•
Lecture 2
Thank you for your attention!