Chapter 2 Flashcards
Functions
What is or a function f(x)
mathematical rule that assigns a real input to output
x=independent/ exogenous
y =dependent/ endogenous variable-
Domain: real numbers
Variables: f,g,h
What is a range?
defines the set of all function values
What is a cost function?
indicates cost level of each production quantity
x = production quantity
y or f(x) = cost
What is a value pair
pair of values describing the variable x & corresponding function value f(x)
What properties does the cartesian coordinate have?
-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
What is a polynomial function
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
What are the relevant polynomial functions in business
constant, linear, quadratic, cubic
What is a constant function?
when polynomial function has a degree of 0
it is used in business for fixed cost function
e.g.
f(x)=c
What is a fixed cost
costs that arise regardless of number of goods produces
e.g. rent payments or loan repayments
What is a linear function?
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
What are quadratic functions?
second degree polynomial
depicts a parabola
e.g. y= ax^2+bx+c
What are properties of parabola
-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
What is the formula for minimum and maximum value of a parabola
x= -b/2a
y= c-b^2/4a
How do you calculate the zeros in a parabola?
By using quadratic formula
What is a slope
provides info how the function changes when value x increases.
If the value increases, then positive
If value decreases, then negative
What can you calculate with a parabola?
Unit cost
What are cubic functions?
Polynomial with third degree
They are real numbers
a cannot be zero
What are cubic functions used for in business math?
complex economic relationships
What are rational functions, give example
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)
What are rational functions used in business math
cost of production
used to determine if company is producing economies of scale- meaning the more it produces the less costs
What is a power function?
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
What is the role of power functions in business?
to illustrate production technology
the contrast of using machines and humans for production
What are the two types of exponential functions?
general exponential functions
natural exponential functions
What are general exponential functions?
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
What roles do exponential and natural functions play in business?
calculation of economic growth. private wealth accounting (development of savings deposits, loan repayments, interests) and valuation of a company’s capital assets (depreciation schedule)
What are natural exponential functions?
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
What are logarithm functions
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
What is a monotonic function?
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
What is a continuous function?
a continuous function has an uninterrupted line
What is a discontinuous function?
If a function has any points of discontinuity.
What are the types of discontinuity points?
Infinity point
Gap
In both cases the function is not continuous
Jump point
the function value jumps with a small change of variable x
What is a piecewise-defined function?
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
What are function compositions?
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
When are function compositions used in business?
to determine the profit function by subtracting cost from sales function
Why is market analysis important?
To determine optimal levels of production and sales for a company to make profit.
Use demand and supply functions to calculate
How do you determine if company is making a profit? What function do you use?
demand and supply function
e.g.
PD (Q)= a- bQ
PS (Q) = a+bQ
PD goes down
PS goes up
Use this to calculate equilibrium
PD=PS
How do you review the success of a company? Which functions do you use?
-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)
What does the price-sales function indicate?
How much a company earns per hour from a sale
What does the revenue function indicate?
the revenue per item sold
What does the cost function indicate?
The cost of production per hour
How to determine if a company is producing optimally?
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
