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FUNCTION(FLOOR FUNCTION) The value of the greatest integer function, denoted by Lx], at the real number x is the greatest integer that is less than or equal to x. EVEN AND ODD FUNCTIONS The function f is an even function if f(-x)=f(x) for all x in the domain of the function. The function f is an odd function if f(-x)= -f(x) for all x in the domain of the function. Types of Function ● C ONE-TO-ONE FUNCTION A function f is a mapping from its domain D to codomain C is said to be one to one(1,-1) or injective if for every x,y in D this holds f(x)= f(y) then x=y. ONTO FUNCTION ● Math A function f is a mapping from domain D to codomain C said to be onto or surjective if for every y element of C there is an element x in domain D such that f(x)= y. Types of Function C Math INVERSE FUNCTION A function f has an inverse function denoted as f-1 if and only if it is a one-to-one function. The graph of f and f-1 the graph of are symmetric with respect to the line y=x J 00- E = mc² Illll Vz LTT DOH H₂0 TIP! FINDING THE INVERSE 1. Make sure f is one-to-one function. 2. Substitute y for f(x). 3. Interchange x and y. 4. Solve for y in terms of x. 5. Substitute £-1 for y. Linear Functions Slope of a Line Slope-Intercept Form of the Equation of a Line General Form of a Linear Equation in Two Variables Point-Slope Form m = Math Y₂ - Y₁ X₂X1 f(x) = mx + b Ax + By = C y-y₁ = m(x-x₂₁) Linear Functions Parallel Lines Perpendicular Lines General Form of a Linear Equation in Two Variables Point-Slope Form c Math m₁ = m₂ 1 m₁ = M2 Ax + By = C y-y₁ = m(x-x₂) Quadratic Functions Quadratic Function A quadratic function can be represented by the equation f(x) = ax² + bx + c, where a, b, and c are real numbers and a # 0. Every quadratic function given by f(x) = ax² + bx + c, a ‡ 0, can be written in standard form as f(x) = a(xh)² + k. The graph of f is a parabola with vertex (h, k). Parabola The graph of a quadratic function given by f(x) = ax² + bx + c₂ a # 0, is a parabola. The coordinates of the vertex of the parabola are b 2a (-2/² √(-2)). The equation of the axis of symmetry is x = 2a The parabola opens up when a > 0 and opens down when a < 0. Minimum or Maximum of a Quadratic Function If a > 0, then the graph of f(x) ax² + bx + c opens up and the vertex (- 12/0² (-2)) √(-2/a) b 2a is the lowest point on the graph; - function. If a < 0, then the graph of f(x) down and the vertex b (-22) 2a is the minimum value of the Math = ax² + bx + c opens (-2/²1 (-2)) is the highest point on the graph; 2a is the maximum value of the function. Exponential and Logarithmic Functions Exponential and Logarithmic Form The exponential form of y = logo x is by The logarithmic form of by = x is y = logb.x. Basic Logarithmic Properties logħb 1 logb1 = 0 C = = logb (bx) = x Math flog₁x = x Exponential and Logarithmic Functions Properties of Logarithms • Product property • Quotient property ● Power property Logarithm-of-each-side property logo (MN) M N logo x = logb = C loga x loga b Math log, M + logo N logo M - logo N logb(MP) = p logh M M = N implies log M = logo N log, M = logo N implies M = N • One-to-one property Change-of-Base Formula If x, a, and b are positive real numbers with a # 1 and b = 1, then Exponential and Logarithmic Functions Exponential and Logarithmic Form The exponential form of y = logo x is by The logarithmic form of by = x is y = logb.x. Basic Logarithmic Properties logħb 1 logb1 = 0 C = = logb (bx) = x Math flog₁x = x ITM 000 00- 0000 E = mc² 21lll Vz VZ ☆ BER TIT DIH H₂O QUESTION FUNCTIONS Find the value of x: In 3x + In 2 = 1 log(log x) = 3 Horizontal Translation of a Graph If f is a function and c is a positive constant, then the graph of Math •y=f(x + c) is a horizontal shift c units left of the graph of y=f(x). •y=f(x - c) is a vertical shift c units right of the graph of y=f(x). Vertical Translation of a Graph Math ● • If f is a function and c is a positive constant, then the graph of • y=f(x) + c is a vertical shift c units upward of the graph of y=f(x). ● • y=f(x) - c is a vertical shift c units downward of the graph of y=f(x). Reflections of a Graph C Math The graph of y = -f(x) is the graph of y=f(x) reflected across the x-axis. y = f(-x) is the graph of y=f(x) reflected across the y-axis. Algebra of Functions c If f and g are functions with domains D and E then 1. (f+g)(x) = f(x) + g(x) 2. (f- g)(x) = f(x) - g(x) Math 3. (f g)(x) = f(x) g(x) 4. (f/g)(x) = f(x) / g(x) where g(x) is not equal to zero 5. (fog)(x) = f(g(x)) J ITM 000 H E = mc² 21lll Vz DIH H₂O Read QUESTION FUNCTIONS Let the domain and codomain of f given below be the real number. f(x) = {x² 1. Graph f. 2. Is f a function? x² + 2x + 1, −1 < x < 1 X, x < -1 3. Is f an odd/even function? 4. What is the range of the f? JT IYM 000 E = mc² 00 21lll Vz Bea DIH H₂O QUESTION FUNCTIONS Let f and g be functions given by, f(x) = x³ + 2x + 2 and g(x) Find: 3(f o g)(x) − (1/3)f(x)g(x) = 2x³+3x²+x+2 3 Synthetic Division Synthetic division is a procedure that can be used to expedite the division of a polynomial by a binomial of the form X - C. ● Math Remainder Theorem Math If a polynomial P(x) is divided by x-c, then the remainder equals P(c). Factor Theorem Math ● A polynomial P(x) has a factor (x - c) if and only if P(c) = 0. Inequalities c Rules for Inequalities Let a, b and c be real numbers. Then the following holds. (1) If a <b, then a + c < b + c. (2) If a < b and c> 0, then ac < bc. (3) If a < b and c < 0, then ac > bc. Note: The inequality is reversed. (4) If a < b and b ≤ c, then a < c. (5) If a < b and a and b have the same sign, then a (6) If 0 < a < b and n is a positive integer, then a" <b" and Va < Vb. Math Terminology Two numbers have the same sign means that both of them are positive or both of them are negative. Remark One common mistake in solving inequalities is to apply a rule with the wrong sign (positive or negative). For example, if c is negative, it would be wrong to apply Rule (2). System of Linear Equations ● ● ● SUBSTITUTION METHOD ELIMINATION METHOD MATRIX (OPTIONAL) Math Sequences and Series C 11.1 Infinite Sequences and Summation Notation ■ Infinite Sequence An infinite sequence is a function whose domain is the set of natural numbers and whose range is a set of real numbers. The terms of a sequence are frequently designated as a1, 92, 93, ans... where an is the value of the function at n. ■n Factorial n factorial, written n!, is the product of the first n natural numbers. That is, n! = 1.2.3 (n-1). n. This is also written in 3.2.1. the reverse order as n! = n(n-1) . Math a1, A2, A3,.... ■nth Partial Sum The nth partial sum of a sequence is the sum of the first n terms of the sequence. The nth partial sum of the sequence n ..,an,... is given in summation notation as Σa₁. i=1 Sequences and Series C = Arithmetic Sequence Let d be a real number. A sequence an is an arithmetic sequence if and only if a;+1 - ai d for all positive integers i. The nth term of an arithmetic sequence is given by an = a₁ + (n − 1)d. Sum of an Arithmetic Series The nth partial sum of an arithmetic sequence (a₁ + a₂), or S₁ = 7/7 [2a₁ + (n − 1)d]. n n an with common difference d is Sn = Math Arithmetic Mean The arithmetic mean of two numbers a and b is a + b 2 Sequences and Series Geometric Sequence Let r be a nonzero real number. The sequence an is ai+1 = r for all i. The nth term of a a geometric sequence if and only if geometric sequence is given by an = a₁rn-1 Math Sum of a Finite Geometric Series The nth partial sum of a geometric sequence a,, with first term a, and common ratio r is S₁ a₁(1-²) 1-r = r # 1. Sequences and Series Math Sum of an Infinite Geometric Sequence The sum of an infinite geometric \r\ < 1. ai sequence an, with first term a, and common ratior is S = 1 Permutation and Combination Math Permutation A permutation is an arrangement of distinct objects in a definite order. The formula for the permutation of n distinct objects taken n! r at a time is P(n,r) (n − r)!* Combination A combination is an arrangement of distinct objects for which the order is not important. The formula for the combination of n n! distinct objects taken at a time is C(n,r) r! (n − r)! = Probability Sample Space An activity with an observable outcome is called an experiment. The sample space of an experiment is the set of all possible outcomes of the experiment. Event An event is a subset of a sample space. Math Probability of an Event Let n(S) and n(E) represent the number of elements, respectively, in the sample space S and the event E. Then the n(E) probability of E is P(E) n(S) Probability C Addition Rules for Probabilities Two events E₁ and E₂ are called mutually exclusive if E₁ E₂ = Ø. If two events are mutually exclusive, then P(E₁ UE₂) = P(E₁) + P(E₂). If two events are not mutually exclusive, then P(E₁ UE₂) = P(E₁) + P(E₂) — P(E₁ E₂). Math Independent Events Two events are independent when the outcome of the first event has no influence on the outcome of the second event. Binomial Probability Formula Let an experiment consist of n independent trials for which the probability of success on a single trial is p and the probability of failure is q = 1 - p. Then the probability of k successes in n trials is given by (1) p²q₁-k. Statistics Math Aside from permutation, combination, and probability, check your companion handbook for the following formulae: 1. Mean 2. Median 3. Mode 4. Range 5. Variance 6. Standard deviation Limits Rules for Limits of Functions at Infinity (L1) lim k = k (where k is a constant) (L2) lim = 0 (where p is a positive constant) (L3) lim = 0 3-∞ bx (L4) lim (f(x) + g(x)) = lim f(x) + lim g(x) The result is valid for sum and difference of finitely many functions. (where b is a constant greater than 1) (LS) lim (f(x) g(x)) = lim f(x) - lim g(x) The result is valid for product of finitely many functions. (L5s) lim (k-g(x)) = k· lim g(x) 818 f(x) x-700 g(x) (L6) lim lim f(x) lim g(x) provided that lim g(x) + 0. Math Limits Rules for Limits of Functions at a Point (Lal) lim k = k (La2) lim x" = a" (La2') lim √x = √a lim √x = √ (La3) lim b* = bª (La4) lim (f(x) + g(x)) = lim f(x) + lim g(x) (where a € R and k is a constant) (where a € R and n is a positive integer) (where a € R and n is an odd positive integer) (where 0 <a € R and n is an even positive integer) (where a € R and b is a positive real number) The result is valid for sum and difference of finitely many functions. (La5) lim (f(x) g(x)) = lim f(x) - lim g(x) 3-0 The result is valid for product of finitely many functions. (La5s) lim (k-g(x)) = klim g(x) (where k is a constant) lim f(x) lim g(x) f(x) (La6) lim x-a g(x) provided that lim g(x) * 0. 214 Math Limits Leading Terms Rule Let f(x) = ax" +a+a₁x+ao and g(x) = bmx +bm-1-1+...+b₁x + bo where an # 0 and bm 0. Then we have f(x) a₁ + a₂-x+.... + a₁x + ao lim x-900 g(x) 10 bmx+bm-x-1+... +bx+bo = lim = Math lim x-00 bm.xm Limits c Sandwich Theorem Let f. g and h be functions such that f(x), g(x) and h(x) are defined for sufficiently large x. Suppose that f(x) ≤ g(x) ≤ h(x) if x is sufficiently large and that both lim f(x) and lim h(x) exist and are equal (with common limit denoted by L). Then we have lim g(x) = L. Remark The condition "f(x) ≤ g(x) ≤ h(x) if x is sufficiently large" means that there is a real number r such that the inequalities are true for all x > r. Example Find lim sin x , if it exists. Explanation The given function can be written as a product of two functions: sinx and. For the second function, its limit at infinity is 0. However, for the first function, its limit at infinity does not exist. Thus we can't apply Rule (5). Solution Since -1 ≤ sin x ≤ 1 for all real numbers x, it follows that sin x 1 75² for all x > 0. Note that lim = lim - = 0. Thus by the Sandwich Theorem, we have lim Math sin x = 0. -0-0- A1²F = mc² (21||| ANSWERING THE QUESTION V2 IDS U DOH H₂O A1²F=mc² IPE V2 Miary + & DIS H₂O JT ITM 000 00- 0000 E = mc² 21llle Vz Bea t RIH H₂O QUESTION What is the sum of the first 200 positive integers? A. 20,000 B. 20,100 C. 21,000 D. 21,100 JT IYM 000 H 0000 E = mc² 21lll Vz TTTT TIT DIA H₂O QUESTION A deck of playing cards consists of 20 cards. How many five-card hands can be chosen from this deck? A. 16,504 B. 1,860,480 C. 15,504 D. 1,680,840 JT ITM 000 0000 E = mc² Illll V2 ARTITS DIH H₂O Reag QUESTION How many real valued x will satisfy the equation 4x²-4x + 3 =0? A. 0 B. 1 C. 2 D. None of the above JT ITM 000 H E = mc² 21llle Vz BER DIH H₂O QUESTION Tyra has test scores of 70 and 81 in her biology class. To receive a C grade, she must obtain an average greater than or equal to 72 but less than 82. What range of test scores on the one remaining test will enable Tyra to get a C for the course? A. [60,90) B. (60,90] C.[65,95) D. (65,95] JT ITM 000 E = mc² 21lll VZ Bea LTT DIAH H₂O QUESTION The blood alcohol concentration (BAC) is the amount of alcohol in a person's blood- stream. A BAC of 0.04% means that a person has 4 parts alcohol per 10,000 parts blood in the body. Relative risk is defined as the likelihood of one event occurring divided by the likelihood of a second event occurring. For example, if an individ- ual with a BAC of 0.02% is 1.4 times as likely to have a car accident as an indi- vidual that has not been drinking, the relative risk of an accident with a BAC of 0.02% is 1.4. Recent medical research suggests that the relative risk R of having an accident while driving a car can be modeled by the equation R = ekx where x is the percent of concentration of alcohol in the bloodstream and k is a constant. J ITM 000 10000 E = mc² 21lll Vz VZ ☆ BER DIH H₂O Medical research indicates that the relative risk of a person having an accident with a BAC is 0.02% is 1.4. Find the constant k in the equation. In 1.4 A. B. C. D. 0.02 ln 0.2 1.4 0.02 QUESTION In 1.4 1.4 ln 0.2