Chapter 2

Question 1

What is a function f(x)

Answer 1

mathematical rule that assigns a real input to output

x=independent/ exogenous
y =dependent/ endogenous variable-

Domain: real numbers
Variables: f,g,h

Question 2

What is a range?

Answer 2

defines the set of all function values

Question 3

What is a cost function?

Answer 3

indicates cost level of each production quantity
x = production quantity
y or f(x) = cost

Question 4

What is a value pair

Answer 4

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

Question 5

What properties does the cartesian coordinate have?

Answer 5

-x abscissa - horizontal
-y ordinate - vertical
-at intersection is origin/zero point
-coordinate has 4 quadrants
2 1
3 4
used to visualize value table
value pairs can be entered

Question 6

What is a polynomial function

Answer 6

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 7

What are the relevant polynomial functions in business

Answer 7

constant, linear, quadratic, cubic

Question 8

What is a constant function?

Answer 8

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

Question 9

What is a fixed cost

Answer 9

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

Question 10

What is a linear function?

Answer 10

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 11

What are quadratic functions?

Answer 11

second degree polynomial
Shape: parabola (u or inverted u)
e.g. y= ax^2+bx+c

Question 12

What are properties of parabola

Answer 12

-if positive, opens up
-if negative, opens down
-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 13

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

Answer 13

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

Question 14

How do you calculate the zeros in a parabola?

Answer 14

By using quadratic formula

Question 15

What is a slope

Answer 15

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

Question 16

What can you calculate with a parabola?

Answer 16

Unit cost

Question 17

What are cubic functions?

Answer 17

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

Question 18

What are cubic functions used for in business math?

Answer 18

complex economic relationships

Question 19

What are rational functions, give example

Answer 19

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 20

What are rational functions used in business math

Answer 20

As a unit cost function
It shows cost of production
used to determine if company is producing economies of scale- meaning the more it produces the less costs

Question 21

What is a power function?

Answer 21

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 22

What is the role of power functions in business?

Answer 22

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

Question 23

What are the two types of exponential functions?

Answer 23

general exponential functions
natural exponential functions

Question 24

What are general exponential functions?

Answer 24

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 25

What roles do exponential and natural functions play in business?

Answer 25

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 26

What are natural exponential functions and what is it’s shape?

Answer 26

function with e 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)= a*e^ c*x

Question 27

What are logarithm functions and what is its shape?

Answer 27

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 28

What is a monotonic function?

Answer 28

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

Question 29

What is a continuous function?

Answer 29

a continuous function has an uninterrupted line

Question 30

What is a discontinuous function?

Answer 30

If a function has any points of discontinuity.

Question 31

What are the types of discontinuity points? 3

Answer 31

-Infinity point -Gap -Jump Jump point - there is a small gap in x value when the line jumps to another y value

Question 32

What is a piecewise-defined function?

Answer 32

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 33

What are function compositions?

Answer 33

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 34

When are function compositions used in business?

Answer 34

to determine the profit function by subtracting cost from sales function

Question 35

Why is market analysis important?

Answer 35

To determine optimal levels of production and sales for a company to make profit. Use demand and supply functions to calculate

Question 36

How do you determine if company is making a profit in the market? What function do you use?

Answer 36

demand and supply function e.g. PD (Q)= a- b*Q PS (Q) = a+b*Q PD goes down PS goes up Use this to calculate equilibrium PD=PS

Question 37

How do you review the success of a company? Which functions do you use?

Answer 37

-price-sales function- linear- downward e.g. p(x) = -mx+b -revenue function- quadratic- downward e.g. R(x) = p(x)*x -cost function -quadratic (s form) upward e.g. C(x) = ax^2+bx+c -profit function-quadratic (sform) down e.g. R(x) - C(x)

Question 38

What does the price-sales function indicate?

Answer 38

How much a company earns per hour from a sale

Question 39

What does the revenue function indicate?

Answer 39

the revenue per item sold

Question 40

What does the cost function indicate?

Answer 40

The cost of production per hour

Question 41

How to determine if a company is producing optimally?

Answer 41

If the maximum point of Production function hits the Revenue function The right side is not optimal because producing too much increases the cost more than revenue The left side is not optimal because you are producing too little, it lowers the profit

Question 42

How do you monitor a company’s success?

Answer 42

Compare sales, revenue, costs

Question 43

What tool can help you analyze sales volume and revenue?

Answer 43

Tabular presentation

Question 44

If you cannot use tabular presentation to analyze sales volume and revenue, what tool should you use?

Answer 44

Math functions in order to describe economic relationships

Question 45

What is a function argument?

Answer 45

The input variable, it’s used to calculate the output or X

Question 46

What is assumed in the domain if there are no restrictions mentioned?

Answer 46

That the domain consists of a set of real numbers or D=R

Question 47

What is a function also called?

Answer 47

Assignment rule

Question 48

What is the symbol of the function value?

Answer 48

y - the value at point x

Question 49

What is function value

Answer 49

The equation which the function value corresponds to point x

Question 50

What is the table of value?

Answer 50

Shows the function value for the specific variable x Helps solve problems involving domain with real numbers

Question 51

How is the cartesian coordinate useful?

Answer 51

Helps visualize the value table

Question 52

What does it mean to a function when the domain corresponds to real numbers?

Answer 52

You can represent and infinite number of value pairs

Question 53

What is the parameter of n in a polynomial function?

Answer 53

zero or natural number

Question 54

What is the parameter of “a” in a any function?

Answer 54

Any real number but it cannot be zero

Question 55

In production, what can variable costs be?

Answer 55

Raw materials

Question 56

What kind if costs is the y-axis intercept?

Answer 56

The fixed cost

Question 57

What type of cost is the variable “a” in a linear function?

Answer 57

Variable costs

Question 58

How many zeros can a parabola have?

Answer 58

1, 2 or no zeros

Question 59

What is a vertex?

Answer 59

An extreme point of a parabola Highest or lowest point of a parabola

Question 60

Why is the unit cost function important in business?

Answer 60

It helps show average costs incurred per unit produced

Question 61

What does the zero in a parabola tell you?

Answer 61

Profit function: break-even point Revenue & cost function: start or end of range/ function

Question 62

What is the general shape of an exponential function?

Answer 62

Always convex Just check if bigger or smaller than 1 to know if + or - And for e or natural expo check if bigger or smaller than 0

Question 63

What functions can either be convex or concave

Answer 63

Power and logarithm functions

Question 64

How does a hyperbola look like?

Answer 64

Like a chair and them mirrored

Question 65

What does e mean?

Answer 65

Eulers number

Question 66

When does the natural logarithm function intersect with the x axis?

Answer 66

when x value is the same as absolute value of parameter a e.g. 2 lnx = it intersects the x axis at 2

Question 67

What is absolute value?

Answer 67

A real number that always corresponds to a positive number eg. The absolute value of -5 and 5 is 5

Question 68

What is the difference between “strictly” monotonically increasing/ decreasing vs monotonically increasing/ decreasing?

Answer 68

The strictly- the slope is always positive or negative (continuously going up/down) The normal one- the slope can be + - OR zero (still going up or down but there may be hindrance)

Question 69

What is a piecewise function?

Answer 69

When a function doesn’t behave continuously and changes characteristics that can be cut into different sections of x