Download pointers in general mathematics and more Lecture notes Mathematics in PDF only on Docsity!
POINTERS IN GENERAL MATHEMATICS
Representing Real-Life Situations Using Functions RELATION - is any set of ordered pairs FUNCTION - is a relation or rule of correspondence between two elements (domain and range) SET OF ORDERED PAIRS A : {(1,2),(2,3),(3,4),(4,5)} B : {(3,3),(4,4),(5,5),(6,6)} TABLE OF VALUES X 1 2 3 4 5 6 Y 2 4 6 8 10 12 ARROW DIAGRAM VERTICAL LINE TEST / GRAPH A relation between two sets of numbers can be illustrated by a graph in the cartesian plane,and that a function passes the vertical line test. The Function Machine Function can be illustrated as a machine where there is the input and the output. PIECEWISE FUNCTION t(m): {250 if 0 < m ≤ 200 ₱250.00 monthly includes 200 free text messages. {(250 + m) if m > 200 ₱1.00 excess message. a b c x y DOM AIN RAN GE JA N DO A NA K E MA N RK M A Y A R E Y
EVALUATING FUNTIONS x = 3 Ex: (^1) ¿ x − 9 2 ¿. x ²+ 4 x − 10 3 ¿ .2 x ²− 6 x + 26 ¿ 3 − (^9) ¿( 3 )²+ 4 ( 3 )− 10 ¿ 2 ( 3 ) ²− 6 ( 3 )+ 26 ¿− 6 ¿ 9 + 12 − 10 ¿ 18 − 18 + 26 ¿ 11 ¿ 26 TYPES OF FUNCTION CONSTANT FUNCTION is a function that has the same output value no matter what your input value is Ex: (^) y = 7 IDENTITY FUNCTION is a function which returns the same value,which was used as its argument. Ex: f ( 2 )= 2 POLYNOMIAL FUNCTION is a function that includes only non negative integer powers. >LINEAR FUNCTION - Function with degree one.It is in the form y=mx+b. Ex: y = 2 x + 5 >QUADRATIC FUNCTION - if the degree of the polynomial function is two,then it is a quadratic function.It is expressed as y=ax²+bx+c,where a ≠ 0 and a,b,c and d are constant and x is a variable. Ex: (^) y = 3 x ²+ 2 x + 5 >CUBIC FUNCTION - Function is a polynomial of degree three. Ex: (^) y = 5 x ³+ 3 x ²+ 2 x + 5 POWER FUNCTION - Is a function in the form (^) y = axb^ where b is any real constant number. Ex: (^) f ( x )= 8 x^5 RATIONAL FUNCTION - is any function which can be represented by a rational function say p ( x ) q ( x ) in which numerator , p(x) and denominator , q(x) are polynomial function of x,where q(x) (^) ≠ 0. Ex: (^) f ( x )= x 2 − 3 x + 2 x 2 − 4 EXPONENTIAL FUNCTION - form of (^) y = abx .If the base b is greater than 1 then the result is exponential growth. Ex: (^) y = 2 x LOGARITHMIC FUNCTION - the inverse of exponential function. Ex: (^) y =log , 49 ABSOLUTE VALUE EQUATION - the non-negative value of x without regard to its sign. Ex: y = 1 x − 41 + 2 GREATEST INTEGER FUNCTION - (always positive) Ex: (^) f ( x )= 11 × 11 + 1 where 11 x 11 is the greatest integer function. OPERATION OF FUNCTION x = 3 Addition ( f + g )( x )= 3 x + 4 f ( x )+ g ( x )= 3 ( 3 )+ 4 ¿ 9 + 4 ¿ 13
P ( 10 )= 10 X − 800
10 X
X = 80 - to break even we need 80 pcs of cookies to be sold d.) (^) P ( x )= 10 x − 800 P ( 250 )= 10 x − 800 250 + 800 = 10 x 1050 10
10 x 10 x = 105 RATIONAL EQUATION - an equation with = symbol involving rational expression. x + 4 x − 1
RATIONAL INEQUALITY - an inequality involving rational expression x + 4 x − 1
RATIONAL FUNCTION - a function of the form f(x)=P(x)/q(x) f ( x )= x + 4 x − 1 FACTORING - CMF 2 x + 2 = 2 ( x + 1 ) x + 2 =( x + 2 ) 4 x 2 − 2 x = 2 x (− x − 1 ) x − 3 =( x − 3 ) PERFECT SQUARE TRINOMIAL x^2 + 2 x + 1 =( x + 1 )( x + 1 ) 4 x 2 − 12 x + 9 =( x − 3 )( x − 3 ) DIFFERENCE OF TWO SQUARE x 2 − 9 =( x + 3 )( x − 3 ) 9 x 2 − 81 =( 3 x + 9 )( 3 x − 9 ) SOLVING RATIONAL EQUATION 2 x
2 x
Step 1: LCD Step 2: Eliminate x = x
x
2 x
2 x = 2 x
x ( 2 )( 3 )( x )−
2 x ( 2 )( 3 )( x )=
( 2 )( 3 )( x )
2 x
12 − 9 = x x = 3 Step 3: Check 2 x
2 x
ADDITION MULTIPLICATION
- plus +¿+¿ liked sign = positive − plus −¿−¿ unlike sign = negative
- plus −¿− plus + ¿ Subtract and copy the sign of Teh bigger one SOLVING RATIONAL INEQUALITIES A. 2 x x + 1
Step 1 Rewrite as single rational Expression and make right side 0(zero) 2 x x + 1
2 x x + 1
2 x x + 1
x + 1 x + 1
x − 1 x + 1
Step 2 Find and check the critical value N : x − 1 = 0 C.V: 1 , − 1 x = 1 0 : x + 1 = 0 x =− 1 Check x − 1 x + 1
Range: (^) { Y ∈ ℜ ㅣ y ≠ 0 } INTERCEPT - one point on ones where the graph intersects X INTERCEPT - point on x-axis where the graph intersects y=0(zeroes) Y INTERCET - point on y-axis where the graph intersects. A. f ( x )= 3 − x x + 1 For x-intercept - y = 0 0 = 3 − x x + 1 let f(x)= 3 − x = 0 x = 3 ∴ (3,0) or x:(3,0) For y-ntercept - x = 0 y = 3 − x x + 1 y =
y = 3 ∴(0,3)^ or y:(0,3) ASYMPTOTE - is an imaginary line to which a graph gets closer and closer as the x or y increases or decreases its value without limit. VERTICAL ASYMPTOTE = DOMAIN HORIZONTAL ASYMPTOTE = RANGE SOLVING EXPONENTIAL EQUATION Exponent - how many times the base will be multiplied by itself. 4 x = 64 4 x = 4 3 x = 3 PROPERTY OF SOLVING EXPONENTIAL EQUATION If (^) bx = bx^^2 then x 1 = x 2 Example Laws 4 x − 1 = 4 (^3) Product Law Example: log 2 6 +^ log 3 M x + 1 = 3 Ⅰ. log (^) b MN =log b M +log b N x = 3 − 1 Quotient Law Example:^ log 3 m n =log 3 M −log 3 N x = 2 Ⅱ.^ log^ b m n =log b M −log b N Power Law Example: (^) log 4 m^3 = 3 log 4 M
Ⅲ. (^) log b M n = nlogb M FRACTION 1 4
− 1 Logarithm Form to Equation Form 1 16
2 =^16
− 2 9 2 = 81 → log 9 81 = 2 3 2 9
2 9 − (^1) ( x − 1 )^2 = 12 → log ( x − 1 ) 12 =^2 Vise Versa log m x = 10 → m 10 = x log 5 ( a − b )= 0 → 5 0 =( a − b ) Example:
2 X + 5
X − 1 Logarithmic Function 16 − 1 ( 2 x + 5 ) = 64 − 1 ( x − 1 ) (^) f ( x )=log 3 x^ −^4 4 2 [− 1 ( 2 x + 5 )] = 4 3 [− 1 ( x − 1 )] (^) Logarithmic Equation 2 [− 1 ( 2 x + 5 )]= 3 [− 1 ( x − 1 )] log 2 4 =log 3 x − 4 x +(− 10 )=− 3 + 3 Logarithmic Inequalities − 4 x + 3 x = 3 + 30 log 3 x < 4 − 1 x − 1
x =− 13 BASIC PROPERTIES AND LAWS OF LOGARITHM PROPERTIES Ex: Ⅰ. log b b = 1 log 3 ³= 1 Ⅱ. log b 1 = 0 log 10 1 = 0 Ⅲ. (^) b log b^ x = x 5 log^510 = 10 Ⅳ. (^) log b bx = x log 7 73 = 3
CELL WALL Protection, structural support and maintenance of cell shape CHLOROPLAST Photosynthesis ENDOPLASMIC RETICULUM Modifies proteins and synthesizes lipids GOLGI BODY Modifies, sorts, tags, packages and distributes lipids and proteins CYTOSKELETON Maintains cell‘s shape, secure organelles on specific positions, allows cytoplasm and vesicles to move within the cell, and enables unicellular organisms to move independently FLAGELLA Cellular locomotion CILIA Cellular locomotion, movement of particles along extracellular surface of plasma membrane, and filtration DIFFERENT TYPES OF MICROSCOPE LIGHT MICROSCOPE (LM) - works by passing visible light through a specimen such as microorganism or a piece of animal or plant tissue. ELECTRON MICROSCOPE (EM) - uses a beam of electrons.The EM has much higher resolving power than light microscope. SCANNING ELECTRON MICROSCOPE (SEM) - uses an electron beam to scan the surface of the cell or group of cells. TRANSMISSION ELECTRON MICROSCOPE (TEM) - is used to study details of internal cell structure. ORGANELLES IN A PLANT CELL ORGANELLES IN AN ANIMAL CELL Cell membrane Cell meambrane Chloroplast Vacuole Vacuole Endoplasmic Reticulum (ER) Endoplasmic Reticulum (ER) Rough ER Rough ER Smooth ER Smooth ER FUNCTION OF CELL MEMBRANE > Barrier > Gatekeeper 4 MAIN COMPONENTS OF THE CELL MEMBRANE > Phospholipids > Carbohydrates > Proteins > Cholesterol WAYS TO TRANSPORT IN CELL MEMBRANE
Endocytosis - when molecules go in Exocytosis - when molecules go out TYPES OF VACUOLE > Contractile Vacuole > Food Vacuole > Sap Vacuole > Gas Vacuole ROUGH ER > lies immediately adjacent to cell nuclei > contains ribosome SMOOTH ER > doesn’t contain ribosome PROKARYOTIC AND EUKARYOTIC CELLS COMMON THINGS TO ALL CELLS > Genetic Material > Cell Membrane > Cytoplasm > Ribosome TWO TYPES OF PLANT TISSUE MERISTEMATIC - cells are capable of cell division PERMANENT - mature cells are incapable of cell division TWO TYPES OF MERISMATIC PLANT TISSUE APICAL - region of cells capable of division and growth in the root and shoot tips of plant LATERAL - helps in increasing the thickness of the plants. TWO TYPES OF PERMANENT PLANT TISSUE SIMPLE - tissue composed of a single cell type COMPLEX - tissue composed of more than one cell type FOUR TYPES OF SIMPLE PERMANENT PLANT TISSUE
