0.1 Number Sets:I. Types of numbers: Tell what type of number fits the description:a. Can be written in the form p/qb. Can be written in the form a+bic. Can be written in decimal notation, but not p/qd. Consists solely of positive and negative whole numberse. Consists of all the numbers which match description “c” and “a”II. Unions and IntersectionsDraw a picture of two circles overlapping. Call one A and one B. Shade in the area of A and B intersecting, and also write A “intersect” B in the appropriate notation. Do the same for the “union” of A and B.III. Distance Formulas:1. The distance between 2 real numbers:2. The distance between two points in a plane (distance formula):3. Midpoint formula 0.2 EquationsThrm: A Product is Zero If and Only If One of the Factors is Zero:If A and B are any real numbers then AB=0 if and only if ___________________Quadratic Formula:Solutions to any quadratic equation ax2 + bx + c = 0 are of the form:Factoring Formulas:For all real numbers a and b,a2-b2=(a-b)(a+b)a3-b3=(a-b)(a2+ab+b2)a2+2ab+b2=(a+b)2a3+3ab2+3a2b+b3=(a+b)3an-bn=(an-1+an-2b+an-3b2+an-4b3+…+a2bn-3+abn-2+bn-1)Problems:1. Solve the equation: 2x5-32x=02. Factor as much as possible: 81x4-13. Find the solution set to the system of equations:x - 2y + z = 0x + y – 3z = 13x – 2y + z = 20.3 Inequalities(Note, ! are used as absolute value signs in the following notes/examples)Theorem 0.10; Algebraic Rules for InequalitiesFor any nonzero real numbers a, b, and c,(a) If a < b and c > 0, then ac < bc(b) If a < b and c < 0, then ac > bc(c) If 0 < a < b, then (1/a)>(1/b)(d) If a < b, then a + c < b + cCase I. When the initial inequality is of the form !ax+b!<c we replace it by:-c < ax+b < cNote that there are essentially 2 inequalities here and the way they are written implies that both first AND the second inequalities must be satisfied. The solution set will then be an INTERSECTION of two setsEx. Solve !3x+7!< 11Case II. When the initial inequality is of the form !ax+b!>c we replace it byax+b< -c OR ax+b>cAgain, we have two inequalities here and the way they are written implies that the first OR the second inequality must be satisfied. The solution set will then be the UNION of these two setsEx: Solve !5x-12! > 8Solving inequalities:1. Find the solutions of the inequality: x 2 + 5x + 6 < 0 x2 – 4x -52. Find the solutions of the inequality: x > x + 6 x + 23. Find the solution of the inequality: x 2 + 1 > 2 x + 20.4 LogicThere are two different types of quantifiers: ______________ and_________________The negation of a statementGiven a statement A, the negation of A, denoted ____________ is a statement that is false whenever A is true, and true whenever A is falseCounterexamplesSuppose the statement “For all x, we have P” is false. A counterexample to this statement is a value of x for which _________________Find the negation to each statement:Statement Negation“ A and B”“A or B”“A  B”“B  A”- Is the statement “Not (A and B) the same as “Not A” and “Not B” ? Why or why not- Is “or” in mathematics terms inclusive or exclusive? ExplainThe Converse of an ImplicationThe converse of the implication A B is the statement:_________________The Contrapositive of an ImplicationThe contrapositive of an implication A B is the statement:_________________0.5 ProofsThe four types of proofs are:1. Calculation2. Direct3. Proof by Contradiction- assume the opposite of what you want to show- show this leads to a contradiction (A and (not A))- this means the assumption was false- what you want to show must actually be true4. Proof by Induction- First show that it is true for: n=____- Next, you are going to assume it is true for n=____- Using the above assumption, you prove for n=____Ex: Use induction to prove that each of the following statements is true for all positive integers n2 + 4 + 6 + …. + 2n = n(n+1)1.1 What is a Function?1. What is a function? Fill in the blanks…Suppose A and B are sets. A function f from A to B is a rule that assigns to _________ element of A a _________ element of B. The set A is called the __________ of f. The set B is called the _________.2. Are these functions?a. b. c.d.e.3. In words, how would you “describe” this: f: R  R? x 1 2 3 4f(x) 4 3 2 1x 1 2 3 4f(x) 1 2 3 2x 1 2 3 2f(x) 4 3 2 1x 1 2 3 4f(x) 4 4 4 4x 1 2 3 4f(x) 1 2 3 44. Fill in the blanks… Given any x that exists in Domain (f), f(x) always represents a _________ element of _________ (f)5. Evaluate this function:If f(x)= x – x2 + 10 , finda. f(2)b.f(-2)c. f(4x + 1)6. In this function a= b(c), What is the independent variable, the dependent variable, and the “name” of the function? 7. Find the domain of these two functions:a. f(x) = (4x2 – 2x)/ SQRT (x2)b. f(x) = SQRT (2x-5))(/ (x2 – x - 2)8. Go back to number 2 now… which of these functions describes the constant function? _________ Which describes the identity function? _________ For the constant function, f(x)= ?____ For the identity function, f(x)= ?____1.2 Graphs of FunctionsFill in the chart for a complete reviewVerbal Description Math Description Draw an example1. y-intercept2. x-intercept3. f is positive4. f is negative5. f is increasing6. f is decreasing7. concave up8. concave down9. f has globalmax10. f has local max1.3 Linear FunctionsLinear functions are generally of the form: ______________Tell which ones are linear: 5y – 2x = 4(1/5) y – (1/2) x = 45y2 – 2x = 4Interpreting slope as a rate of changeIf f(x) = mx + b then a unit change in the ___________________ variable always produces a change in the ____________________ variableAverage Rate of Change of a Function on an Interval-Give the definitionDo linear functions have a constant rate of change on a specific interval?_______ If so, what must it always be equal to? ________. In your above definition of a linear function, what does m represent?___________ b?____________A proportional function can be written in the form:________________. What letter describes the proportionality constant?______Equations for lines: Name Info Given Eq of lineSlope intercept form slope m, y intercept b _________________Point-slope form _________________ y-y0 = m(x-x0)_______________ (x0, y0) & (x1, y1) _________________Describe the relationship between the slopes when lines are parallel? What about perpendicular?Ex: Find the linear function that is