2. Functions and Graphs
2.1 Functions
Cartesian Coordinate System
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
,
exactly one y function (rule for finding y)
For every input, there is exactly one output.
Rule: Compute y by dividing x by 5
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
The symbol
For any element x in the domain of the function f, the symbol represents an element in the range of f , which corresponds to x in the domain of f. If x is an input value, then is the corresponding output value. If x is not in the domain of f, then f is not defined at x, and does not exist.
Domains?
Applications
Cost function
Cost=(fixed costs)+(variable costs)
Price-Demand function
Revenue function
Revenue=(number sold) * (price per item)
Profit function
A camera manufacturer wholesales to retail outlets across the US. The company produced price-demand data per the following table:
2 / 87
5 / 68
8 / 53
12 / 37
The company then modeled the data to get the price-demand function ,
2.2 Elementary Functions: Graphs and Transformations
Library of Elementary Functions (pictures on p.60)
Vertical and horizontal shifts
Reflections, Stretches, and Shrinks
Graph Transformations
Vertical translation
Horizontal translation
Reflection
Vertical expansion/contraction
Piecewise defined functions
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 price-demand 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.
2.3 Quadratic Functions
If a, b, and c are real numbers with , then the function 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 in the rectangular coordinate system, and find its intercepts.
Solve the quadratic inequality graphically and symbolically.
How many intercepts are possible?
Vertex form:
Given the quadratic function
Find the vertex form for f.
Find the max/min of the function. State the range.
Discuss the relationship between and .
A camera manufacturer wholesales to retail outlets across the US. The company modeled supply and demand data to get the price-demand function , . The revenue function is therefore . What price will maximize revenue?
Given production costs at , 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:
A polynomial function is a function that can be written in the form
for n, a non-negative integer, called the degree of the polynomial. The coefficients, are real numbers with . 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 turning points and can cross the x axis at most n times.
Polynomial Root Approximation
(skip)
If r is a zero of the polynomial
then
.
Approximate (to four decimal places) the real zeros of
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 / 44y / 5 / 12 / 26 / 56 / 96 / 152 / 226 / 326 / 536
Rational Functions
A rational function is any function that can be written in the form , where and are polynomials and . The domain is the set of all real numbers such that .
Find the domain and the intercepts for the rational function
Graph
(skip)
Finding vertical and horizontal asymptotes
Let be a rational function in lowest terms.
To find a vertical asymptote, solve for x. If is a real number such that , then is a vertical asymptote of the graph of . (if a is a zero of both and , then is not in lowest terms. Factor out from both.
Horizontal asymptote (divide by highest pwr of x)
1) If the degree of is less than the degree of , is a horizontal asymptote.
2) If the degree of is equal to the degree of , then is a horizontal asymptote, where a and b are the leading coefficients of and .
3) If the degree of is greater than the degree of , there is no horizontal asymptote.
Graphing Rational Functions
Given the rational function
Find intercepts and equations for any vertical or horizontal asymptotes
Using this information and additional points as necessary, sketch a graph of f for and .
Find asymptotes
A company that manufactures computers has established that, on the average, a new employee can assemble components per day after t days of on-the-job training, as given by , . Sketch a graph of N, , including any vertical or horizontal asymptotes.
2.5 Exponential Functions
Power function:
Exponential fcn:
A function f represented by where , 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.
is a reflection of
Basic Properties of ,
1. All graphs will pass through the point .
2. All graphs are continuous curves with no holes or jumps.
3. The x-axis is a horizontal asymptote.
4. If , then increases as x increases.
5. If , then decreases as x increases.
Sketch a graph of ,
Properties of Exponential Functions
Natural Exponentiation
For calculation purposes, assume one puts $1 in a savings account for 1 year at 100% interest.
Compounding /m
Annual / 1 / 2Monthly / 12 / 2.613035
Daily / 365 / 2.714567
Hourly / 8760 / 2.718127
Minutely / 525,600 / 2.718279
Secondly / 31,536,000 / 2.718282
Continuous
Euler’s number – irrational, like
- natural exponentiation
Calculator exercise
Exponential functions with base e are defined by
Cholera bacteria multiplies exponentially by cell division as given approximately by , where N is the number of bacteria present after t hours and 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 . 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
If $1000 is invested at 10%, compounded monthly, how much will be in the account after 10 years?
Continuous compounded interest
Starting with P=100, r=0.08, and t=2 years, examine m as m increases without bound.
Compounding /m
/ AAnnual / 1 / 116.64
Semiannually / 2 / 116.9859
Quarterly / 4 / 117.1659
Weekly / 52 / 117.3367
Daily / 365 / 117.349
Hourly / 8760 / 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 .
If $1000 is invested at 10%, compounded continuously, how much will be in the account after 10 years?
Simple Interest:
Compound Interest:
Continuously compounded interest:
2.6 Logarithmic Functions
Inverse Functions
Reversible actions Gal to pints
Pts to gal
A B C … X Y Z
C D E … Z A B HELP JGNR
Open door, get in, close door, start engine
Shut off eng, open door, get out, close door
Notation
% time skies cloudy in Augusta GA
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 1243 / 40 / 39 / 29 / 28 / 26 / 27 / 25 / 30 / 26 / 31 / 39
A function is a one-to-one function if, for elements c and d in the domain of f,
implies
(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 is on the graph of f, then is on the graph of .
Let . Find and verify inverse func
Logarithmic Functions
The inverse of an exponential function is called a logarithmic function.
Common Logarithms
The common logarithm of a positive number x, denoted , is defined by
if and only if
where k is a real number. The function given by
is called the common logarithm function.
Logarithms with other bases
The logarithm with base a of a positive number x, denoted , is defined by
if and only if
where , and k is a real number. The function given by
is called the logarithmic function with base a.
(a logarithm is an exponent)
If for some k, then .
Natural logarithms
If for some k, then .
John Napier (1550-1617)
Calculator exercise
Properties of Logarithmic functions
For positive numbers m, n, and ,
1.
2.
3.
4.
5.
6.
7.
8.
Change of Base formula
Let x, , and be positive real numbers. Then,
How long (to the next whole year) will it take money to double if it is invested at 20% compounded annually?
Compound Interest:
3. Mathematics of Finance
3.1 Simple Interest
Interest on a loan of $100 at 12% for 9 months
Amount: Simple Interest
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%?
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?
$0-2499 / $29+1.6%
$2500-9999 / $49+0.8%
10,000+ / $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?