2. Functions and Graphs
2.1 Functions
Cartesian Coordinate System
y  9  x2
II
I
III
IV
A function is a rule (process or method) that
produces a correspondence between two sets of
elements such that to each element in the first set,
there is corresponds one and only one element in
the second set. The first set is called the domain
and the second set is called the range.
5
1
a
w
A
x
10
2
b
x
B
y
15
3
y
C
y  x x  1,
x 1
x
f
f(x)
exactly one y  function (rule for finding y)
For every input, there is exactly one output.
x
Rule: Compute y by dividing x by 5
f x  x
5
4 y  3x  8
y 2  x2  9
x2  y 4
Vertical line test
An equation defines a function if each vertical
line in the coordinate system passes through at
most one point in the graph of the equation.
If any vertical line passes through two or more
points on the graph of the equation, then the
equation does not define a function.
If a function is specified by an equation and the
domain is not indicated, then we assume the
domain is the set of all real number
replacements of the input variable that produce
real values for the output variable. In many
applied problems, the domain is determined by
practical considerations within the problem.
meaningful input – domain
corresponding output – range
Finding the Domain
y
1
x
y  4 x
y
1
4 x
The symbol f  x 
For any element x in the domain of the function
f, the symbol f  x  represents an element in the
range of f , which corresponds to x in the
domain of f. If x is an input value, then f  x  is
the corresponding output value. If x is not in the
domain of f, then f is not defined at x, and f  x 
does not exist.
f x 
12
x2
g x  1  x2
h x   x  1
f 6, f 2, g 2, f  2, f 0  g 1  h10
Domains?
f x  x2  2x  7
f a 
f a  h
f a  h  f a 
f a  h   f a 
h
Applications
Cost function
Cost=(fixed costs)+(variable costs)
C  a  bx
Price-Demand function
p  m  nx
Revenue function
Revenue=(number sold) * (price per item)
Profit function
P  R C
A camera manufacturer wholesales to retail
outlets across the US. The company produced
price-demand data per the following table:
x(millions)
2
5
8
12
p($)
87
68
53
37
The company then modeled the data to get the
price-demand function p x   94.8  5x ,
1  x  15
2.2 Elementary Functions: Graphs and
Transformations
f x  x2
g x  x2  4 
h x    x  4 
2
k  x   4x 2 
Library of Elementary Functions (pictures on p.60)
f x  x
h x   x 2
m x   x 3
n x   x
p x   3 x
g x  x
x  64
x  12.75
Vertical and horizontal shifts
f x  x
f x  x  4
f x  x  5
f x  x  4
f x  x  5
Reflections, Stretches, and Shrinks
f x  x
f x  2 x
f  x   0.5 x
f x   x
f  x   2 x
Graph Transformations
Vertical translation
k  0
y  f x  k 
k  0
Horizontal translation
h  0
y  f  x  h 
h  0
Reflection
y   f x
Vertical expansion/contraction
A  1
y  Af  x 
0  A  1
y   x  3 1
Piecewise defined functions
 x , x  0
x 
 x, x  0
Utilities
Easton Utilities uses the rates from the table
below to compute each customer’s monthly
natural gas bill.
$0.7866 per CCF for the first 5 CCF
$0.4601 per CCF for the next 35 CCF
$0.2508 per CCF for all over 40 CCF
Price-Demand (skip to 2.3)
At the beginning of the 21st century, the world
demand for crude oil was about 75 million barrels
per day and the price of a barrel fluctuated between
$20 and $40. Suppose the daily demand for crude
oil is 76.1 million barrels when the price is $25.52
per barrel and the demand drops to 74.9 million
barrels when the price rises to $33.68. Assuming a
linear relationship between the demand x and the
price p, find a linear function that models the pricedemand relationship for crude oil. Use this to
predict the demand if the price rises to $39.12.
The daily supply of crude oil also varies with the
price. Suppose that the daily supply is 73.4 million
barrels when the price is $23.84, and this supply
rises to 77.4 million barrels when the price rises to
$34.24. Assuming a linear relationship between
supply x and price p, find a linear function that
models the price-supply relationship for crude oil.
Use this model to predict the supply if the price
drops to $20.98 per barrel.
The price tends to stabilize at the point of
intersection of the demand and supply functions.
This point is called the equilibrium point.
2.6 x  167  6.8 x  543
2.3 Quadratic Functions
If a, b, and c are real numbers with a  0 , then
the function f  x   ax 2  bx  c is a quadratic
function and its graph is a parabola.
Solution methods:
Square root property
Factoring
Quadratic Formula
Completing the Square
Sketch a graph of f  x    x 2  5 x  3 in the
rectangular coordinate system, and find its
intercepts.
Solve the quadratic inequality  x 2  5 x  3  0
graphically and symbolically.
How many intercepts are possible?
Vertex form: f  x   a x  h  k
f  x   2 x 2  16 x  24
Given the quadratic function f  x   0.5x 2  6 x  21
Find the vertex form for f.
Find the max/min of the function. State the range.
Discuss the relationship between f  x  and
g x  x2.
A camera manufacturer wholesales to retail
outlets across the US. The company modeled
supply and demand data to get the price-demand
function p x   94.8  5x , 1  x  15 . The
revenue function is therefore
R x   xp x   x94.8  5x . What price will
maximize revenue?
Given production costs at C  x   156  19.7 x ,
what amount of cameras will maximize profit?
What is the wholesale price that will maximize
profit? Where are our break-even points?
2.4 Polynomial and Rational Functions
Constant fcn:
Linear function:
Quadratic function:
Cubic function:
f x  b
f  x   mx  b
f  x   ax 2  bx  c
f  x   ax3  bx 2  cx  d
A polynomial function is a function that can be
written in the form
f  x   an x n    a2 x 2  a1 x  a0
for n, a non-negative integer, called the degree
of the polynomial. The coefficients,
an , an 1 , a1 , a0 are real numbers with an  0 .
The domain of a polynomial function is the set
of all real numbers.
Turning points and x intercepts of Polynomials
(skip)
The graph of a polynomial function of positive
degree can have at most n  1 turning points and
can cross the x axis at most n times.
Polynomial Root Approximation
(skip)
If r is a zero of the polynomial
P x   x n  an1 x n1  an2 x n2    a1 x  a0
then
r  1  max an1 , an2  a1 , a0 .
Approximate (to four decimal places) the real
zeros of P x   2 x 4  5 x 3  4 x 2  3x  6
Using the length of a fish to estimate its weight
is of interest to both scientists and sport anglers.
The data in the table gives the average weights
of lake trout for certain lengths. Find a
polynomial model that can be used to find the
weights of lake trout for certain lengths.
x 10 14 18 22 26 30 34 38 44
5 12 26 56 96 152 226 326 536
y
Rational Functions
A rational function is any function that can be
n x 
written in the form f  x  
, where n x  and
d x
d  x are polynomials and d  x   0 . The domain
is the set of all real numbers such that d  x   0 .
Find the domain and the intercepts for the
rational function
3x
f x  2
x 4
3x
Graph f  x   2
x 4
(skip)
Finding vertical and horizontal asymptotes
n x 
Let f  x  
be a rational function in lowest terms.
d x
To find a vertical asymptote, solve d  x   0 for
x. If a is a real number such that d  x   0 , then
x  a is a vertical asymptote of the graph of
y  f  x . (if a is a zero of both n x  and d  x ,
then f  x  is not in lowest terms. Factor out
 x  a  from both.
Horizontal asymptote (divide by highest pwr of x)
If the degree of n x  is less than the degree
d  x , y  0 is a horizontal asymptote.
If the degree of n x  is equal to the degree
d  x , then y  a is a horizontal asymptote,
b
where a and b are the leading coefficients of
n x  and d  x .
3) If the degree of n x  is greater than the
degree of d  x , there is no horizontal asymptote.
1)
of
2)
of
Graphing Rational Functions
3x
x2  4
Find intercepts and equations for any vertical or
horizontal asymptotes
Given the rational function f  x  
Using this information and additional points as
necessary, sketch a graph of f for  7  x  7 and
 7  y  7.
x 1
Find asymptotes 2
x 1
A company that manufactures computers has
established that, on the average, a new
employee can assemble N t  components per
day after t days of on-the-job training, as given
50t
by N t  
, t  0 . Sketch a graph of N,
t4
0  t  100 , including any vertical or horizontal
asymptotes.
2.5 Exponential Functions
Power function: f  x   x 2
Exponential fcn: g  x   2 x
A function f represented by f  x   b x where
b  0, b  1, is an exponential function with base
b. The domain of f is the set of all real numbers
and the range of f is the set of all positive real
numbers.
2 x is a reflection of 2  x
Basic Properties of f  x   b x , b  0, b  1
1. All graphs will pass through the point 0,1.
2. All graphs are continuous curves with no
holes or jumps.
3. The x-axis is a horizontal asymptote.
4. If b  1, then b x increases as x increases.
5. If 0  b  1, then b x decreases as x increases.
Sketch a graph of f  x   12 4 x ,  2  x  2
Properties of Exponential Functions
a x  a y  a x y
ax
x y

a
ay
x y
a   a x y
x
ab   a xb x
x
x
a
a
 
   x
b b
ax  a y  x  y
ax  bx  a  b
Natural Exponentiation
For calculation purposes, assume one puts $1 in
a savings account for 1 year at 100% interest.
Compounding
Annual
Monthly
Daily
Hourly
Minutely
Secondly
Continuous
m
1
12
365
8760
525,600
31,536,000
A1
2
2.613035
2.714567
2.718127
2.718279
2.718282
Euler’s number – irrational, like 
e  2.718281828459...
f  x   e x - natural exponentiation
Calculator exercise
e1
e 0.5
e  2.56
Exponential functions with base e are defined by
y  ex
y  e x
Cholera bacteria multiplies exponentially by cell
division as given approximately by N  N 0e1.386 t ,
where N is the number of bacteria present after t
hours and N 0 is the number initially present. If
we start with 25 bacteria, how many bacteria will
be present
In 0.6 hour?
In 3.5 hours?
Cosmic ray bombardment of the atmosphere
produces neutrons, which in turn react woth
nitrogen to produce radioactive Carbon-14.
Carbon-14 enters all living tissues through CO2,
which is first absorbed by plants. Carbon-14 is
maintained at a constant level until the organism
dies, at which point, it decays according to
A  A0e 0.000124 t . If 500 mg is present in a sample
from a skull at the time of death, how much is
present after 15,000 years? After 45,000 years?
The half-life of Carbon-14 is the time t at which
the amount present is ½ the original amount.
Find the half-life of Carbon-14.
Compound Interest
If P (present value) dollars is invested an annual
rate of interest r, compounded m times per year,
then after t years, the account will contain A
(future value) dollars, where
mt
r


A  P 1  
 m
If $1000 is invested at 10%, compounded
monthly, how much will be in the account after
10 years?
Continuous compounded interest
 r
A  P 1  
 m
mt
Starting with P=100, r=0.08, and t=2 years,
examine m as m increases without bound.
Compounding
Annual
Semiannually
Quarterly
Weekly
Daily
Hourly
m
1
2
4
52
365
8760
A
116.64
116.9859
117.1659
117.3367
117.349
117.351
If a principal P is invested at an annual rate r
compounded continuously, then the amount A at
the end of t years is given by A  Pe rt .
If $1000 is invested at 10%, compounded
continuously, how much will be in the account
after 10 years?
Simple Interest: A  P1  rt 
mt
r


Compound Interest: A  P1  
 m
Continuously compounded interest: A  Pe rt
2.6 Logarithmic Functions
Inverse Functions
Reversible actions f  x   8 x Gal to pints
g  x   f 1  x   x 8 Pts to gal
ABC…XYZ
CDE…ZAB
HELP  JGNR
Open door, get in, close door, start engine
Shut off eng, open door, get out, close door
x7
2
2x  7
Notation
x 1
f 1  x 
% time skies cloudy in Augusta GA
1 2 3 4 5 6 7 8 9 10 11 12
43 40 39 29 28 26 27 25 30 26 31 39
f 3 
f 1 

A function is a one-to-one function if, for
elements c and d in the domain of f,
c  d implies f c   f d 
(different inputs result in different outputs)
If f is a one-to-one function, then the inverse of
f is the function formed by interchanging the
independent and dependent variables for f. If
a, b  is on the graph of f, then b, a  is on the
graph of f 1  x .
f :a b
f 1 : b  a
Let f  x   x3  2 . Find and verify inverse func
Logarithmic Functions
The inverse of an exponential function is called
a logarithmic function.
Common Logarithms
g  x   log x
x  10k  log x  log10k  k
The common logarithm of a positive number x,
denoted log x , is defined by
log x  k if and only if x  10k
where k is a real number. The function given by
f  x   log x
is called the common logarithm function.
log1 
1
log

1000
log 10 
Y1  10  X
Y2  log X
Logarithms with other bases
a 1
The logarithm with base a of a positive number
x, denoted log a x , is defined by
log a x  k if and only if x  a k
where a  0, a  1, and k is a real number. The
function given by
f  x   log a x
is called the logarithmic function with base a.
(a logarithm is an exponent)
If x  2k for some k, then log 2 x  k .
Natural logarithms
If x  e k for some k, then ln x  k .
John Napier (1550-1617)
Calculator exercise
log 3,184
log5 25  2
1
log 9 3 
2
1
log 2    2
 4
ln 0.000349
log 3.24
43  64
6  36
1
2 
8
3
y  log 4 16
log 2 x  3
y  log8 4
logb 100  2
Properties of Logarithmic functions
For positive numbers m, n, and a  1, r  
logb 1  0
logb b  1
logb b x  x
b log b x  x
logb M  logb N  logb MN
M
6. log b M  log b N  log b
N
7. logb M p   p logb M
8. logb M  logb N  M  N
1.
2.
3.
4.
5.
wx
logb

yz
logb wx  5 
3
e x ln b 
ln x

ln b
3
2
logb 4  logb 8  logb 2  logb x
2
3
log x  log x  1  log 6
log x  2.315
ln x  2.386
10 x  2
ex  3
3x  4
Change of Base formula
Let x, a  1, and b  1 be positive real numbers. Then,
log a x 
log b x
log b a
e x ln b  b x
log5 38.25 
How long (to the next whole year) will it take
money to double if it is invested at 20%
compounded annually?
 r
Compound Interest: A  P1  
 m
mt
3. Mathematics of Finance
3.1 Simple Interest
I  P  r t
Interest on a loan of $100 at 12% for 9 months
Amount: Simple Interest
A  P  P  r t
 P1  rt 
Amount due on a loan of $800 at 9% for 4 months
Present value of an investment
How much should you pay to get $5000 in 9
months at 10%?
At   P  P  r  t
If you buy a 180-day treasury bill with maturity
value of $10,000 for 9,893.78, what rate of
interest is earned?
You finance the sale of your car at 10% over
270 days, for $3500. 60 days later, you sell the
note for $3550. What rate of interest will the
buyer get?
Transaction size
$0-2499
$2500-9999
10,000+
Commission
$29+1.6%
$49+0.8%
$99+0.3%
An investor purchases 50 shares of stock at
$47.52/share. After 200 days, the investor sells
the stock for $52.19/share. Using the
commission table, find the annual rate of
interest earned.
A credit card has an annual interest rate of
21.99%, and simple interest is calculated by the
average daily balance. In a 30-day billing cycle,
purchases of $56.75, $184.36, and $49.19 were
made on days 12, 19, and 24, respectively. A
payment of $100 was made on day 10. If the
unpaid balance at the start of the billing cycle
was $842.67, how much interest will be charged
at end of cycle, and what will the next unpaid
balance be?
3.2 Compound and Continuous Compound Interest
A  P1  rt 
$1000 deposited at 8%, compounded quarterly
A1 
A2 
A3 
A4 
An  A0 1  r 
n
 r
An  A0 1  
 m
mn
A  P1  i 
$1000 deposited at 8% over 5 years
Compound interest for various periods
Annually
n
A  P1  i 
n
Semiannually
r
n
A  P1  i 
i
m
Quarterly
n
A  P1  i 
Monthly
n
A  P1  i 
 r
A  P 1  
 m
mt
A  P1  i 
n
mt
m
  rt
r
 r   P 1  r 
 1
A  P 1  
 P 1  


 m
 m
 x
x rt
 1  
 P 1     Pe rt
 x  
xrt
$1000 deposited at 8% over 5 years
compounded continuously.
$5000 invested @ 8% for 2 years compounded:
daily?
continuously?
Present Value
How much should you invest now at 10%
compounded quarterly to have $8000 toward the
purchase of a car in 5 years? Continuously?
Growth Rate
A recent growth oriented mutual fund has
grown from $10,000 to $126,000 in the last 10
years. What interest rate, compounded
annually, would produce the same growth?
compounded continuously?
Growth time
How long will it take $10,000 to grow to
$12,000 if it is invested at 9% compounded
monthly?
Y1  100001.0075
n
Y2  12000
A  P1 I ^ N =0
Find the APYs for each of these banks.
Annual Percentage Yield
Bank
Rate
Compounded
Advanta
4.93
monthly
DeepGreen
4.95
daily
CharterOne
4.97
quarterly
Liberty
4.94
continuously
Which CD has the best return?
If a principal is invested at the annual (nominal)
rate r compounded m times a year, then the
annual percentage yield is
m
r


APY  1    1
 m
The annual percentage yield is also referred to
as the effective rate or true interest rate.
Computing the Annual Nominal Rate Given the
Effective Rate
A savings and loan wants to offer a CD with a
monthly compounding rate that has an effective
rate of 7.5%. What annual nominal rate
compounded monthly should they use?
3.3 Future Value of an Annuity; Sinking Funds
Ordinary Annuity - payments made at the end of
each time interval
Future Value - sum of all the payments made
plus all interest earned.
$100 deposit every 6mo, 6% semiannually, 3 yrs
S ni 
1  i n  1
i
Future Value of an Ordinary Annuity
1  i n  1
FV  PMT
 PMTsni
i
What is the value of an annuity at the end of 20
years if $2,000 is deposited each year into an
account earning 8.5% compounded annually?
How much of this value is interest?
Sinking Funds
Computing the Payment for a Sinking Fund
A company estimates that it will have to replace
a piece of equipment at a cost of $800,000 in 5
years. To have this money available in 5 years,
a sinking fund is established by making equal
monthly payments into an account paying 6.6%
compounded monthly. How much should the
payment be? How much interest is earned
during the last year?
Growth in an IRA
Jane deposits $2000 annually into a Roth IRA
that earns 6.85% compounded annually. Due to
a change in employment, these deposits stop
after 10 years, but the account continues to earn
interest until Jane retires 25 years after the last
deposit. How much is in the account at Jane's
retirement?
Approximating Interest Rates
A person makes monthly deposits of $100 into
an ordinary annuity. After 30 years, the annuity
is worth $160,000. What annual rate
compounded monthly has this annuity earned
during this 30-year period?
3.4 Present Value of an Annuity; Amortization
How much should you deposit in an account
paying 6% compounded semiannually in order
to be able to withdraw $1000 every 6 months
for the next 3 years?
Present Value of an Ordinary Annuity
n
1  1  i 
PV  PMT
i
What is the present value of an annuity that pays
$200 per month for 5 years if money is worth
6% compounded monthly?
Retirement Planning
Recently, Lincoln Benefit Life offered an
ordinary annuity that earned 6.5% compounded
annually. A person plans to make equal annual
deposits into this account for 25 years in order
to make 20 equal annual withdrawals of
$25,000. How much must be deposited
annually to accumulate sufficient funds to
provide for these payments? How much total
interest is earned during this entire 45 year
process?
Amortization
Monthly Payment and Total Interest on an
Amortized Debt
You buy a TV set for $800 over 18 mo at 1½ %
monthly. What is the payment? How much
interest will you pay?
Amortization Schedules
If you borrow $500 over 6mo at 1% monthly,
how much is interest and how much is
principal?
Equity in a home
A family purchased a home 10 years ago for
$80,000. The home was financed by paying
20% down and signing a 30 year mortgage at
9%. The market value of the house is now
$120,000. How much equity does the family
have?
Automobile financing
You have negotiated a price of $25,200 for a
new pickup. Options are 0% financing for 48
months, or a $3000 rebate (loan at 4.5%
compounded monthly).
Problem Solving Strategy for Finance Problems
1. Determine whether the problem involves a single
payment or a sequence of periodic payments.
a. Simple/Compounded interest problems have
a single present and single future value.
b.Annuities may be concerned with a
present/future value, but involve a sequence
of payments.
2. Single payment - simple/compound interest?
a. Simple interest is usually used for
durations of a year or less
b. Compound interest for longer periods.
3. Sequence of payments
a. increasing in value? - future value problem
b. decreasing value? - present value problem
(amortization)
Interest:
I  P  r t
Simple Interest:
A  P1  rt 
Compound Interest:
A  P1  i 
n
1  i   1
n
Future value of an
ordinary annuity:
FV  PMT
Present value of an
ordinary annuity:
1  1  i 
PV  PMT
i
i
n