• You will need to refer to chapter 1 for some terminology and concept information
• Open your Pre-Calc book to page 152. We are going to see how to translate typical equationsolving to “if-then” statements or “if-and-only-if” statements
• Now turn to page 43 to look at the Law of
Transitivity in the green box

• Reversible steps correspond to if-and-only if statements (a.k.a. biconditional statements)
• Nonreversible steps correspond to if-then statements for which the converse is false (i.e. squaring both sides of the equation from page
152)
• There are lots of steps that are always reversible
(partial list on page 153)
• These rules enforce the idea that you must check each solution in the original equation to make sure you don’t include any extraneous solutions
• Ex1. Solve x

14 x 2

• Ex2.
5 4 n

3
• Ex3. log
 t
3 
3 t
2
5 t 7


3log
 t

1


• This book typically write g(f(x)) as g ◦ f(x)
• The value of x is in the domain of g ◦ f if and only if x is in the domain of f (which allows f to be performed) and f(x) is in the domain of g (which allows g to be performed)
• Ex1. Suppose f(x) = x³ + 8 and g(x) = 4x – 5 and both are defined for all real numbers x. Find g ◦ f
• The composition of functions is NOT commutative

• Remember that inverse functions are reflections over the line y = x
• A function has an inverse if and only if it is a 1-1 function
• Test for inverses: f(g(x)) = x AND g(f(x)) = x
5
• Ex2. Solve x 7

243

• Open your book to page 167 where we will look at the blue box near the top of the page
• When asked to give some properties of a function be sure to be descriptive and use math terms
• Include things like: domain, range, symmetry (if applicable), relative extrema, end behavior, and asymptotes (if applicable)
• Ex1. For all x, let h be defined by h(x) = x²·cos x.
Give some properties of h.

• Chunking was originally covered in Algebra I
• Chunking is the practice of thinking of an expression as if it were a single variable
• Ex1. What would be the chunk in the equation e
4 x 
13 e
2 x 
36
• Ex2. Solve the equation from Ex1.
• Ex3. Find all real solutions to ( m

5)
4  
• Ex4. Solve y
8 
8 y
5 
12 y
2 
0 m

• To find the intersection points of two functions you could graph and use the intersection function, or you could subtract one function from the other, graph the difference, and look for the zeros of the difference function
• Open your book to page 177 to look at the graph
• If two functions are each continuous, so is their sum, difference, and product

• If f is a continuous function on the interval a < x < b and if one of the values f(a) or f(b) is positive while the other is negative, then f has at least one zero between a and b
• Open your book to page 180 to look at example 2
• Ex1. Use the Intermediate Value Theorem to show that there must be a solution to the equation sin x = .4·ln x between 2 and 3
• If you keep applying the Intermediate Value
Theorem to smaller and smaller intervals, you can more precisely estimate the zeros of a function

• Another way to estimate the zeros of a function accurately is called the Bisection Method
• The Bisection Method basically involves bisecting an interval by testing the midpoint of that interval thereby cutting the possibilities in half (see page
181)
• The zero function on your graphing calculator can also estimate the zeros of a function, but you need to know how to use these other methods

• Solving inequalities are a little more difficult than solving equations because you have to be cognizant of the sense of the inequality
• Open your book to page 184 and read the theorem in the green box
• Ex1. Find all real numbers x that satisfy 5 x 
7
• If g is an increasing function on an interval, then x such that f(x) and h(x) are in that interval,
( )

( )

( ( ))

( ( ))

• If g is a decreasing function on an interval, then x such that f(x) and h(x) are in that interval,
( )

( )

( ( ))

( ( ))
• Ex2. Can the sum of a number and that number raised to the sixth power ever be negative?

• Another way to solve complex inequalities is to graph the two sides of the inequality separately and write your solution from the graph (you will have to use the intersect and zero functions on your graphing calculator)
• Open your book to page 192 and look at the orange box
• Ex1. Find the solutions of the inequality
5x² + 6x – 23 < -x² + 5x + 12

• Open your book to page 193 (Example 3)
• Ex2. Apply the Test Point Method to solve the inequality 8 x
4 
4 x
3 
17 x
2 
2 x
4 
3 x
3 
18 x
2

• Remember that absolute value equations may have two answers but some are extraneous
• Ex1. Solve │2x – 10│ = 3x
• You can graph and check the intersection
• The distance between two points on a number line is often called the absolute error of the points
• Distance between two points: │x
1
– x
2
│
• The theorem on page 197 shows you how to change an absolute value inequality into a double inequality

• Absolute value is frequently used in quality control situations in the real world
• Ex2. If the absolute error can be no more than
.04 cm for oversize circular washers with a hole of diameter 1.4 cm, write inequalities that an acceptable diameter d must satisfy.
a) Using absolute value b) Without using absolute value
•
• Open your book to page 198 (example 3)
5 n

2
Ex3. The sequence has the terms
2 n

3
7 12 17 22 27
, , , , ,...
5 7 9 11 13 is b n and . For what values of n n lim
 b n

2.5
within .01 of its limit?

• The Graph Translation Theorems are restated on page 203 of the PreCalculus book (green box)
• Formula for a circle with radius r and center (h, k) is given by (x – h)² + (y – k)² = r²
• Most of this section is a review from earlier this year
• Ex1. Find all maximum or minimum points and symmetries of the graph of the function with equation y = 3│x – 2│ – 4

• Consider the following function: ( )
    o This is the difference of the identity and the floor functions o The graph is on page 210 o What remains for positive x is the part to the right of the decimal point o There is a key on your graphing calculator for this o fPart is found by: Math, NUM, 4
• Open your book to page 211 where we are going to look at the parabola information at the bottom of the page and on to the next page