Gen Math Reviewer PDF

GEN MATH REVIEWER APPLICATIONS OF EXPONENTIAL EQUATION Exponential growth- example: bacteria doubles every hour Compound interest Population growth EXPONENTIAL FUNCTION example: f(x)= 3x Note: Bawal maging negative ang base. EXPONENTIAL EQUATION...

GEN MATH REVIEWER APPLICATIONS OF EXPONENTIAL EQUATION Exponential growth- example: bacteria doubles every hour Compound interest Population growth EXPONENTIAL FUNCTION example: f(x)= 3x Note: Bawal maging negative ang base. EXPONENTIAL EQUATION example: 3X+1 = 24 EXPONENTIAL INEQUALITY example: 1/8x > 5 ▪ SOLVING EXPONENTIAL EQUATION - Both sides should have the same base Example: 3x+2 = 27 3x+2 = 33 x+2=3 x=3–2 x=1 ▪ SOLVING INEQUALITY - REVERSE/FLIP INEQUALITY SIGN –> dividing negative both sides by a negative number - RETAIN SIGN -> base is greater than 1 Example: 53x > 253x-2 53x > 52(3x-2) 3x > 2(3x-2) (get the exponent) 3x > 6x – 4 3x-6x > -4 -3x > -4 -3x/-3 > -4/-3(divide both sides by -3) x < 4/3 (reverse inequality sign since we divide both sides by -3) Answer: {𝑥 ϵ ℝ | 𝑥 < 4/3} ▪ GRAPH OF EXPONENTIAL FUNCTION ▪ f(x) = 2x x -2 -1 0 1 2 f(x) 1/4 1/2 1 2 4 o Substitute x values --→ 2-2 , 2-1 , 20 , 21 , 22 ▪ f(x) = 3x f(2) = 32 = 9 f(3) = 33 = 27 EXPONENTIAL FUNCTION ▪ DOMAIN: ALL INTEGERS : {𝑥 ϵ ℝ } ▪ RANGE: ONLY THE POSITIVE OUTPUTS : {𝑦 ϵ ℝ | 𝑦 > 0} ▪ X-INTERCEPT : NONE ▪ Y-INTERCEPT: (0, 1) ▪ ZERO: NONE ▪ HORIZONTAL ASYMPTOTE: y = 0 ▪ VERTICAL ASYMPTOTE: NONE ❖ A population of bacteria doubles every hour. If the initial population is 10, 0000 bacteria, after how many hours will the population exceed 640,000? Solution: 10,000 (2)t/1 = 640,000 10,000 (2)4/1= 160,000 10,000 (2)5/1= 320,000 10,000 (2)6/1= 640,000 → Therefore, the answer is 6. ❖ The predicted population of LUNHS is given by P = 1000 e (0.10 y), where y represents the number of years after 2024. What will be the population in the year 2034? Solution: P = 1000 e (0.10 y) y= 2034 – 2024 = 10 P = 1000 e (0.10)(10) P = 2,718 Note: round off if needed ❖ Suppose the half-life of a certain radioactive substance is 40 seconds, and the initial amount is 20 grams. What exponential model represents the remaining amount of the substance over time? 1 Answer: y = 20 (2)t/40 Determine the amount of substance remaining after 2 minutes. (convert 2 minutes into seconds= 2x60= 120) 1 t/40 Solution: y = 20 (2) 1 y = 20 (2)120/40 y = 2.5 LOGARITHMIC FUNCTION f(x) = logb x Note: base= subscript CONVERT LOGARITHMIC EQUATION TO EXPONENTIAL EQUATION AND VICE VERSA ▪ log3 4 = a → 3a = 4 ▪ 5x = 3 → log53 = x APPLICATIONS OF LOGARITHMIC FUNCTION 1. Earthquake magnitude 2. Acidity Level 3. Sound Intensity LOGARITHMIC FUNCTION Example: y = log3 4 (x-2) LOGARITHMIC EQUATION Example: log3 8 = log3 4 (x-2) LOGARITHMIC INEQUALITY Example: log3 4 (x-2) > 9 ❖ SOLVING LOGARITHMIC EQUATION Example: log3 (x – 2) = 3 33 = x - 2 27 = x – 2 27 + 2 = x 29 = x ❖ SOLVING LOGARITHMIC INEQUALITY Example: log3 (x – 3) > 2 32 > x – 3 9>x–3 9+3>x 12 > x or x < 12 Answer: {𝑥 ϵ ℝ | 𝑥 < 12} What is the first step in solving logarithmic inequality such as (x – 2) > 5 Answer: Add 2 to both sides. (Additive inverse) ❖ PROPERTIES OF LOGARITHM o PRODUCT PROPERTY Examples: log3 8 + log3 4 → log3 (8 4) log3 (8 4) → log3 8 + log3 4 o QUOTIENT PROPERTY Examples: log3 8 - log3 4 → log3 (8/4) log3 (8/4) → log3 8 - log3 4 f(x) log3 x o Substitute x values. x 1 3 9 27 f(x) 0 1 2 3 GRAPH OF f(x) = logb x 1. No horizontal asymptote 2. Vertical asymptote : x = 0 3. Domain: Set of all possible input values 4. Range : Set of all real numbers GRAPH OF LOGARITHMIC FUNCTION ▪ DOMAIN: ALL INTEGERS : {𝑥 ϵ ℝ | 𝑥 > 0} ▪ RANGE: ONLY THE POSITIVE OUTPUTS : {𝑦 ϵ ℝ } ▪ X-INTERCEPT : (1,0) ▪ Y-INTERCEPT: NONE ▪ ZERO: x=1 ▪ HORIZONTAL ASYMPTOTE: NONE ▪ VERTICAL ASYMPTOTE: x = 0 Note: Inverse lang sila ng exponential equation, magkapalit lang sila. ❖ Suppose that an earthquake released approximately 𝟏𝟎𝟏𝟐 joules of energy. What is its magnitude on a Richter scale? ❖ The decibel level of sound in a quiet office is 𝟏𝟎−𝟔 watts/𝐦𝟐. What is the corresponding sound intensity in decibels? ❖ A 1-liter solution contains 10-5 moles of hydrogen ions. Find its pHlevel. Note: INPUT LANG SA CALCULATOR Magreview para hindi maging asymptotic to pasado ang score mo, yung malapit sa passing score pero never ma-meet ang passing score! Okay lang kahit ganyan kay crush, huwag lang sa grade. Eyy! GOODLUCK, GRADE 11! Love, Ma’am Ganda