1. Read Module 3 Topic 7, Module 4 Topic 1 and 2, Module 5 Topic 1~2.
2. Do the drills for the topics.
3. Read the Chapter 3 sections 2, 5 and Chapter 4 sections 1~3 in your textbook.
4. Do Homework for week 5 (you can find the list in the conference).

Week 5 Supplementary Notes
Chapter 3

Section 3.2: Polynomial Function of Higher Degree
A polynomial function P is given by , where the coefficients are real numbers and the exponents are whole numbers. This polynomial is of nth degree.

Far-Left and Far-Right Behavior
The behavior of the graph of a polynomial function as x becomes very large or very small is referred to as the end behavior of the graph. The leading term of a polynomial function determines its end behavior.

x becomes very large x → ∞ x becomes very large x → ∞ x becomes very small
-∞ ← x x becomes very small
-∞ ← x

We can summarize the end behavior as follows:

The Leading-Term Test

If is the leading term of a polynomial, then the behavior of the graph as x → ∞ or as x → −∞ can be described in one of the four following ways.

If n is even and an >0:

▼ ▼ | If n is even and an <0:▲ ▲ |

If n is odd and an >0: ▲▼ |

If n is odd and an <0: ▲

▼ |

Polynomial Function, Real Zeros, Graphs, and Factors (x − c)
If c is a real zero of a function (that is, f(c)=0), then (c,0) is an x-intercept of the graph of the function.

Some higher degree polynomials can be factored by grouping
Example: Find the zeros of
Solution: We factor by grouping as follows:

Then, by the principle of zero products, the solutions of the equation are 2, -3, and 3. These are the zeros of.

Example: Find the zeros of
Solution: We factor as follows:

We now solve the equation to determine the zeros. We use the principle of zero products: or or or These are the zeros of

Note: Only the real-number zeros of a function corresponds to the x-intercepts of its graph. For instance, the real-number zeros of the function in the above example, only and can be seen on the graph of the function but the non-real zeros and - cannot been seen on the graph.

Even and Odd powers of (x − c)
If (x−c)k , k≥1, is a factor of a polynomial function P and: * k is even, then the graph then the graph intersects but does not cross the x-axis at (c,0) * k is odd, then the graph crossed the x-axis at (c,0)

To graph a polynomial function: 1) Use the leading-term test to determine the end behavior of a polynomial function. 2) Find the zeros of the function by solving. Any real zeros are the first coordinates of the x-intercepts. 3) Use the x-intercepts (zeros) to divide the x-axis into intervals and choose a test point in each interval to determine the sign of all function values in that interval. 4) Find. This gives the y-intercept of the function. 5) If necessary, find additional function values to determine the general shpe of the graph and then draw the graph. 6) As a partial check, use the facts that the graph has at most n, x-intercepts and at most n−1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or does cross but touch the x-axis.

Section 3.5: Graphs of Rational Functions and Their Applications

Given that and are polynomials and . * is a rational function where domain of is all x such that * If and , then f is discontinuous (undefined) at . * If , and , then is an x-intercept for the graph of .

Note: We assume that all the rational functions we consider are in the lowest terms unless otherwise stated.

Vertical and Horizontal Asymptotes:
Horizontal and Vertical Asymptotes * A graph of a function never crosses or touches vertical asymptotes. * A graph of a function may cross or touch horizontal asymptotes but as or , the function approached a certain y-value

Easier interpretation
Vertical Asymptote:
Let x = 5 be a vertical asymptote for a rational function, then the graph of the function will never cross or touch the vertical line, x = 5.

Horizontal Asymptote:
Let y = 3 be a horizontal asymptote for a rational function, then the graph of the function may touch or cross the horizontal line, y = 3, in some places, but the behavior of the graph approach the horizontal asymptote, y = 3, as x goes to either positive or negative infinity.

Horizontal Asymptotes and Rational function
Easier interpretation:
Let be a rational function where the numerator function is an degree polynomial and the denominator function is an degree polynomial.
Then,
1. for m < n, the line y = o is a horizontal asymptote. 2. for m = n, the line y = am / bn is a horizontal asymptote. 3. for m > n, there is no horizontal asymptote.

Chapter 4

Section 4.1: Inverse Functions
Recall, a function is a set of ordered pairs with the property that no two ordered pairs have the same 1st component and different 2nd component (i.e. Functions can have ordered pairs with different 1st component and same 2nd component).

Inverse Function:
If the ordered pairs of a function g are the ordered pairs of a function f with the order of the coordinated reversed, then g is the inverse function of f.

One-to-One Function
A function f is one-to-one if different inputs have different outputs—that is, if, then .
Or a function f is one-to-one if when the outputs are the same, the inputs are the same—that is, ,then .

Condition for an Inverse Function:
A function f has an inverse function if and only if f is one- to one.

Graph of Inverse Functions (p336):
The graph of function and its’ inverse function are symmetrical with respect to the line .

What does it mean for a function to be One-to-One?
If a function is one-to-one, then its inverse correspondence is a function. Another words, if f(x) is a function, its inverse function exists.
If f is one-to-one, then exists (i.e. the inverse function exists and it is also one-to-one).

How to determine if a function is one-to-one? 1) Horizontal Line test 2) To verify not one-to-one, just find 2 ordered pairs with different 1st component and same 2nd component.

Composition of Inverse Function Property:
A function f is one-to-one function, then is the inverse function of f if and only if for all x in the domain of and for all x in the domain of f.

To find the Inverse of a Function f
Step 1) Verify that f is one-to-one. If not one-to-one, then does not exist.
Step 2) Replace f(x) by y.
Step 3) Interchange x and y.
Step 4) Solve for y.
Step 5) Replace y by

Example: Find for
Solution:
Step 1) f is one-to-one
Step 2)
Step 3)
Step 4)

Step 5)

Section 4.2: Exponential Functions and Their Applications
Exponential Function
The exponential function with the base b is defined by , where and x is a real number.

Graphs Exponential Functions,, and their properties: * The function f is a one-to-one function. It has the set of real numbers as its domain and the set of positive real numbers as its range. * The graph of f is smooth, continuous curve with a y-intercept of (0,1) and the graph passes through (1, b).

, where | , where |

Exponential Function with Base e (Natural Exponential function)
For all real numbers x, the function defined by is called the natural exponential function.

A calculator can be used to evaluate ex for specific values of x. For example, e = e1 = 2.718281… , e2 = 7.389056… , e3.5 = 33.114552… , e-1.4 = 0.246597…
Remember, “e” is a real number. It is an irrational number, but real number.

Section 4.3: Logarithmic Functions and Their Applications
Recall the exponential function . Since exponential functions are one-to-one, their inverses exist. We call the inverses “Logarithmic Functions” or “Log functions” for short.

For the function, the inverse function is .
“” read “the logarithm (or log), base 2, of x” means “the power to which we raise 2 to get x.”

Logarithmic Function:
If x > 0 and b is a positive constant (b≠1), then if and only if .

The function defined by is a logarithmic function with base b. This function is the inverse of the exponential function.

Know how to convert from Log form to exponential form and vice versa; A logarithm is an exponent!

Of all possible logarithmic bases, the base “e” and the base “10” are used most exclusively.

Notation for Common Logarithm (logarithm with base 10) is “log” and the notation for Natural Logarithm (logarithm with base e) is “ln”.

Logarithmic Notation:
Common logarithm:
Natural logarithm:

Learn and Know how to evaluate the logarithms of base 10 and base e using your calculator

Logarithmic-Exponential Relationships: is equivalent to is equivalent to

Basic Logarithmic Properties: 1. 2. 3. 4.

Graphs Logarithmic Functions, , and their properties: * The function f is a one-to-one function. It has the set of positive numbers as its domain and the set of all real numbers as its range. * The graph of f is smooth, continuous curve with a x-intercept of (1,0) and the graph passes through (b, 1).

, where | , where |