Chapter 2

Question 1

What is f(x)?

Answer 1

value of the function at the location x
can also be symbolized as: y
y is called dependent variable-because value depends on variable x
y is called endogenous variable- because it s determined by model

Question 2

What is a function?

Answer 2

mathematical rule that assigns a unique real number to each real number x
x is called the function argument
x is independent variable- because numbers are selected independently from domain.
x is also called exogenous variable- because it is determined outside model

the domain is real numbers
is always specified by terms
symbols used: f,g,h

Question 3

What is a range?

Answer 3

defines the set of all function values

Question 4

What is a cost function?

Answer 4

indicates cost level of each production quality
x (independent variable)- represents production quantity
y or f(x) (dependent variable)- represents the cost

Question 5

What is a value pair

Answer 5

pair of values describing the variable x & corresponding function value f(x)

Question 6

What properties does the cartesian coordinate have?

Answer 6

-x plotted horizontal number line (abscissa)
-y or f(x) plotted in vertical number line
(ordinate)
-at the intersection lies the origin (zero point) of coordinates system
-coordinate has 4 quadrants
2 1
3 4
used to visualize value table
value pairs can be entered

Question 7

What is a polynomial function

Answer 7

is a function that is defined by its terms, which are formed through the addition or subtraction of powers
e.g.
y=anx^n+ an-1x^n-1
n cannot be zero

Question 8

What are the relevant polynomial functions in business

Answer 8

constant, linear, quadratic, cubic

Question 9

What is a constant function?

Answer 9

when polynomial function has a degree of 0
it is used in business for fixed cost function
e.g.
f(x)=c

Question 10

What is a fixed cost

Answer 10

costs that arise regardless of number of goods produces
e.g. rent payments or loan repayments

Question 11

What is a linear function?

Answer 11

first degree polynomial
used to depict costs in business (variable & fixed)
e.g.
y= ax+b
a cannot be zero both a and be are real numbers
a is slope (y2-y1/x2-x1)
b is y-axis intercept

Question 12

What are quadratic functions?

Answer 12

second degree polynomial
depicts a parabola
e.g. y= ax^2+bx+c

Question 13

What are properties of parabola

Answer 13

-if positive, opens up
-if negative, opens down
-can have 1, 2 or no zeros
-has an extreme point or vertex
-lowest point is called minimum point
-value of x which parabola reaches lowest is called minimum
-if the vertex is highest point its called maximum point
-value of x which parabola reaches highest is called maximum

Question 14

What is the formula for minimum and maximum value of a parabola

Answer 14

x= -b/2a
y= c-b^2/4a

Question 15

How do you calculate the zeros in a parabola?

Answer 15

By using quadratic formula

Question 16

What is a slope

Answer 16

provides info how the function changes when value x increases.
If the value increases, then positive
If value decreases, then negative

Question 17

What can you calculate with a parabola?

Answer 17

Unit cost

Question 18

What are cubic functions?

Answer 18

Polynomial with third degree
They are real numbers
a cannot be zero

Question 19

What are cubic functions used for in business math?

Answer 19

complex economic relationships

Question 20

What are rational functions, give example

Answer 20

Quotient of two polynomial functions
denominator cannot be zero- meaning graph approaches y axis, but never touches or intersects it
e.g:
f(x) = h(x)/ g(x)

Question 21

What are rational functions used in business math

Answer 21

cost of production
used to determine if company is producing economies of scale- meaning the more it produces the less costs

Question 22

What is a power function?

Answer 22

a, b, x can be any real number, but a can’t be zero and x needs to be positive
If b >1, then it has positive slope. The graph is convex- graph is steeper as x increases
if b is <1, then it has a negative slope. The graph is convex, but becomes flatter as x increases
If 0<b<1 then it has positive slope. The graph is concave (cuba) it becomes flatter as x increases
e.g.
f(x) =ax^b

Question 23

What is the role of power functions in business?

Answer 23

to illustrate production technology
the contrast of using machines and humans for production

Question 24

What are the two types of exponential functions?

Answer 24

general exponential functions
natural exponential functions

Question 25

What are general exponential functions?

Answer 25

b is called base and it has to be a positive real number/
a and c can be any real number but c can’t be zero.
x is an exponent!!!
if 0<b<1 then it has a negative slope and is convex. y intersection is a and doesn’t touch x axis (no zero)
if b is >1 slope is positive and is also convex. y intersection is a and doesn’t touch x axis (no zero)
e.g.
f( x) = ab ^cx

Question 26

What roles do exponential and natural functions play in business?

Answer 26

calculation of economic growth. private wealth accounting (development of savings deposits, loan repayments, interests) and valuation of a company’s capital assets (depreciation schedule)

Question 27

What are natural exponential functions?

Answer 27

e is a constant called euler’s number
a and c can be any real number, but c can’t be zero and a has to be positive.
If c<0 then it has negative slope and is convex. a is the intersection in y axis. it doesn’t touch x axis (no zero).
If c>0 then it has positive slope and is convex. a is the intersection in y axis. it doesn’t touch x axis (no zero).

e.g.
f(x)= ae^ cx

Question 28

What are logarithm functions

Answer 28

there are natural and general logarithm functions
x can only be positive real numbers. a can be any real number except zero.
When the function intersects with the x axis it is absolute value of a.
If a>0, then it has positive slope and is concave.
If a <0 then graph has negative slope and is convex
e.g. f(x) = a ln x

Question 29

What is a monotonic function?

Answer 29

a function that either increases or decreases with increasing x-values
The slope doesn’t change.

Different properties of monotonic functions:

-Function is strictly monotonically increasing, if the function values increases while x increases, the slope is positive
-Function is monotonically increasing, if the function values increases while x increases or remains constant, the slope is positive or zero
-Function is strictly monotonically decreasing, if the function values decreases while x increases, the slope is negative
-Function is monotonically decreasing, if the function values decreases while x increases or remains constant, the slope is negative or zero

Question 30

What is a continuous function?

Answer 30

a continuous function has an uninterrupted line

Question 31

What is a discontinuous function?

Answer 31

If a function has any points of discontinuity.

Question 32

What are the types of discontinuity points?

Answer 32

Infinity point
Gap
In both cases the function is not continuous

Jump point
the function value jumps with a small change of variable x

Question 33

What is a piecewise-defined function?

Answer 33

functions defined in sections

properties:
can have a vertical jump using vertical solid line
having a jump point means functions is not continuous.
it can change from a straight lines into parabolas

Question 34

What are function compositions?

Answer 34

If two functions h and g have identical domains they can generate a new function f by adding, subtracting or multiplying

If two functions h and g have identical domains and h cannot be zero they can generate a new function f by
dividing both functions

It two functions are concatenated or composed the can be represented as outer and inner functions

Question 35

When are function compositions used in business?

Answer 35

to determine the profit function by subtracting cost from sales function