GenMath-Q1_W1-ABC (1) PDF

EXERCISE Direction: Read the problem carefully and answer on a sheet of paper. General...

EXERCISE Direction: Read the problem carefully and answer on a sheet of paper. General Math Cadbury 9 chocolate bar costs Php12.00 per piece. However, if you buy more than 20 pieces, they will be marked down to a price of Php10.00 per piece. Use a piecewise function to represent the cost in terms of number of chocolate bars bought. Numeracy Skill Directions: Read the problem carefully. Write your answer on a sheet of paper. Represent the given situation as a function in various forms. Quarter 1 Week 1 - A 1. A person is earning Php390.00 per day to do a certain job. (M11GM-Ia-1) Express the total salary (S) as a function of the number of (n) days The learner represents real-life situations using that the person works. functions, including piece-wise functions. 2. A taxi ride costs Php 40.00 as flag-down rate and each Writer: JOHNLEX L. DIVINAGRACIA additional meter (or a fraction thereof) adds Php4.00 to the fare. Use a Layout Artist: function to represent the taxi fare in terms of distance d in meters. Key in or click the link to learn more: KRISHEA MAE P. JARUDA https://youtu.be/oeYUd5SFJys EPS I - Mathematics KIM S. ARCEÑA, EdD  GENERALIZATION EXAMPLE A relation is any set of one or more ordered pairs. A relation can also The temperature of an object outside the refrigerator x is different from its be a rule that relates the domain (set of all first coordinates of the temperature inside the refrigerator f(x). Suppose, the function 𝑓 𝑥 = 𝑥 2 + ordered pair) and the range (set of all second coordinates of the ordered 3𝑥 − 4 represents the temperature, in degrees, of a certain object inside a pair). refrigerator. Express the temperature of the object inside the refrigerator in An ordered pair consists of two components,(x, y), where x and y are various forms for 𝑥 = −1, 0, 1 any real number. SOLUTION A function is a special type of relation where each element in the domain is paired to the range only once by some rule. Also, all Tabular Form functions are relations, but not all relations are functions. x -1 0 1 A function may be represented as: 1. Ordered Pair f(x) -6 -4 0 2. Table 3. Arrow/Mapping Diagram Ordered pair: { −1, −6 , 0, −4 , 1, 0 } Equation: f 𝑥 = 𝑥 2 + 3𝑥 − 4 4. Equation or Formula; and Mapping: Graph: 5. Function as a Set of Ordered Pairs(𝒙, 𝒇 𝒙 ) 6. Graphically Function as a Set of Ordered Pairs(𝒙, 𝒇 𝒙 ) A function is a set ordered pair (x, y), where the x-value appears only once in the set, otherwise the it is not a function. The set of all x − values is called the domain. The set of all y − valuesis called the range. EXERCISE Function described by a Table Directions: Read and understand. Write your solutions on a sheet of All the points in the ordered pair can be tabulated using two rows; the paper. independent variable and the dependent variable, to show their relationship. Express 𝑦 = 𝑥 2 + 2𝑥 + 3 for 𝑥 = −4, 0, 4 in various forms. Function expressed in Arrow/Mapping Piecewise Function In the tabular form, the possible values of x and y are arranged These are functions which are defined in different domains since they are chronologically and lines are drawn from the independent variable to determined by several equations. their corresponding dependent variable. EXAMPLE Function as an Equation or Formula A user is charged Php2,000.00 monthly for a mobile plan which includes an The function can be expressed as 𝑦 = 𝑓(𝑥), where the dependent ePhone 6s Plus, unlimited same network texts, 16 gb data allowance monthly variable 𝑦 𝑜𝑟 𝑓(𝑥) is written on the left side and the independent variable and a 100 texts to other networks. Messages in excess of 100 are charged x is written at the right side of the equation. Php1.00 each. Represent the monthly cost for text messaging using the function t(m), where m is the number of messages sent in a month. Functions described Graphically Through inspection, the graph of a relation can easily be determined if it SOLUTION is a function or not, it is with the use of Vertical Line Test. To do this, The cost of messaging can be expressed by the piecewise function: simply draw a vertical line passing through the graph of the relation in as many possible points of intersection. If the vertical line intersects the graph at exactly one point, then the relation is a function. Otherwise, it is a mere relation. t(m) =  2 000 , if 0 < m < 100 2 000 + (m – 100) , if m > 100  General Math Directions: Read each item carefully. Write your solutions on a sheet of paper. 1. Solve: Given 𝑓 𝑥 = 𝑥 3 + 𝑥 2 + 3𝑥 − 12. a. Solve for: 𝑓(3) 𝑓(7) 𝑓(𝑥 + 1) 𝑓(𝑥 2 − 1) b. Is 𝑓(𝑥 + 1) the same as 𝑓(𝑥 − 2)? Quarter 1 Week 1– B 2. ML computer shop charges Php30.00 per hour (or a fraction of (M11GM-Ia-2) an hour) for the first two hours and additional of Php15.00 per The learner evaluates a function. hour for each succeeding hour. Find how much would you pay if you used one of their computers for: Writer: a. 45 minutes JOHNLEX L. DIVINAGRACIA b. 4 hours Layout Artist: c. 155 minutes KRISHEA MAE P. JARUDA EPS I - Mathematics KIM S. ARCEÑA, EdD Key in or click the link to learn more: https://youtu.be/wvFeAVWHo_Q  GENERALIZATION EXERCISES 1 Directions: Read each item carefully. Write your answer on a In the notation f(x), it is stressed that f is the name of the function sheet of paper. and f(x) is the value of f at x. If y is the value of f at x or y = f(x), x is Evaluate the following functions. called the independent variable since any element of the domain can 1. 𝑔 𝑥 = 𝑥 − 3, when x = 3 replace it, and y is called the dependent variable because its value 2. ℎ 𝑥 = 𝑥 2 − 3𝑥 + 5, when x = -5 depend on the value of the independent variable x. To evaluate 𝑥 2 −1 3. 𝑖 𝑥 = , when x = -0.6 𝑥−1 functions, just replace all the values of x in the equation and simplify. 𝑥 3 −8 4. 𝑗 𝑥 = , when x = a + b 𝑥−2 EXAMPLES 5. 𝑘 𝑥 = 𝑥 4 − 2𝑥 2 + 2, when x = 𝑥 2 Let f(x) = 2x – 3. Find: a. 𝑓(0) EXAMPLE b. 𝑓 5 The velocity V (m/s) of a ball thrown upward after time t seconds is c. 𝑓(−1) given by V(t) = 20 – 9.8t. What is the velocity of the ball after 1 d. 𝑓(0.5) second? after 2 seconds? e. 𝑓(𝑚) f. 𝑓(𝑥 − 1) SOLUTION g. 𝑓 𝑥 2 𝑉 1 = 20 − 9.8 1 = 20 − 9.8 = 10.2 𝑉 2 = 20 − 9.8 2 = 20 − 19.6 = 0.4 h. 𝑓(𝑥 2 − 1) After 1 second, the ball’s velocity is 10.2 m/s. After 2 SOLUTION seconds, its velocity is 0.4 m/s. It means as time passes by, the a. To find f(0), replace the value x in 𝑓 𝑥 = 2𝑥 – 3 with 0. In other velocity of the ball is constantly decreasing. words, simply substitute 0 for all x. 𝑓 0 = 2 0 − 3 = 0 – 3 = −3 EXERCISES 2 b. 𝑓 5 = 2 5 − 3 = 10 − 3 = 7 c. 𝑓 1 = 2 −1 − 3 = −2 − 3 = −5 Directions: Read the problem carefully. Write your answer on a d. 𝑓 0.5 = 2 0.5 − 3 = 1 − 3 = −2 sheet of paper. e. 𝑓 𝑚 = 2 𝑚 − 3 = 2𝑚 − 3 f. 𝑓 𝑥 − 1 = 2 𝑥 − 1 − 3 = 2𝑥 − 2 − 3 = 2𝑥 − 5 The profit made by a manufacturing company producing bags is given g.𝑓 𝑥 2 = 2 𝑥 2 − 3 = 2𝑥 2 − 3 by the function f 𝑥 = 3𝑥 + 250 , where 𝑥 is the number of bags sold. h. 𝑓(𝑥 2 − 1) = 2(𝑥 2 − 1) − 3 = 2𝑥 2 − 2 − 3 = 2𝑥 2 − 5 How much is the profit of the company if they sold 300 pieces of bags? How many bags are produced if they have a profit of Php2,014.00?  EXERCISES General Directions: Read each item carefully. Write your solutions on a Math sheet of paper. Let 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 𝑎𝑛𝑑 𝑔 𝑥 = 4𝑥 2 − 1. Find: a. 𝑓 + 𝑔 𝑥 𝑏. 𝑓 – 𝑔 𝑥 c. 𝑓 𝑔 𝑥 𝑑. (𝑓 / 𝑔) (𝑥) e. 𝑓 ◦ 𝑔 𝑥 𝑓. 𝑔 ◦ 𝑓 𝑥 Numeracy Skill Directions: Read each item carefully. Write your solutions on a sheet of paper. Quarter 1 Week 1– C (M11GM-Ia-3) Let 𝑓 𝑥 = 𝑥 3 − 1 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 2 − 1. The learner performs addition, subtraction, Find: multiplication, division and composition of a. 𝑓 + 𝑔 𝑥 functions. 𝑏. (𝑓 − 𝑔) (𝑥) c.(𝑓 𝑔) (𝑥) Writer: d. (𝑓 / 𝑔) (𝑥) JOHNLEX L. DIVINAGRACIA e. 𝑓 ◦ 𝑔 𝑥 Layout Artist: KRISHEA MAE P. JARUDA 𝑓. 𝑔 ◦ 𝑓 𝑥 EPS I - Mathematics Key in or click the link to learn more: KIM S. ARCEÑA, EdD https://youtu.be/3gaxVHVI4cI  GENERALIZATION Product of Functions Operation on Functions Let 𝑓 𝑥 = 2𝑥 2 + 3 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 2 − 2𝑥. If f and g are functions, their sum is defined as : Find (𝑓 𝑔) (𝑥) (𝑓 + 𝑔) (𝑥) = 𝑓(𝑥) + 𝑔(𝑥) (𝑓 𝑔) (𝑥) = (2𝑥 2 + 3) (𝑥 2 − 2𝑥) Multiply = 2𝑥 4 − 4𝑥 3 + 3𝑥 2 − 6𝑥 Simplify if possible. If f and g are functions, their difference is defined as : (𝑓 − 𝑔) (𝑥) = 𝑓(𝑥) − 𝑔(𝑥) Quotient of Functions If f and g are functions, their product is defined as : Let 𝑓 𝑥 = 4𝑥 2 − 1 𝑎𝑛𝑑 𝑔 𝑥 = 2𝑥 2 + 3𝑥 + 1. 𝑓 𝑔 𝑥 = 𝑓(𝑥) 𝑔(𝑥) Find (f / g) (x). 4𝑥 2 −1 If f and g are functions, their quotient is defined as : (𝑓 / 𝑔) (𝑥) = 2𝑥 2 +3𝑥+1 𝑓 𝑓 𝑥 (2𝑥−1)(2𝑥+1) 𝑔 𝑥 = 𝑔 𝑥 = Express in factored form if possible (2𝑥+1)(𝑥+1) If f and g are functions, the composite function is defined as : 2𝑥+1 = Simplify 𝑥+1 (𝑓 ◦ 𝑔) (𝑥) = 𝑓[𝑔(𝑥)] EXAMPLES Function Composition. The symbol 𝑓 ◦ 𝑔 is read as “f circle g” or “f of g”. It means that in computing (f ◦ g) (x), first, apply the function f to Sum of Functions x and then the function f to g(x). Let 𝑓 𝑥 = 3𝑥 2 − 4𝑥 + 5 𝑎𝑛𝑑 𝑔 𝑥 = 2𝑥 3 + 6𝑥 − 2. Let 𝑓 𝑥 = 4𝑥 2 − 5𝑥 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 + 1. Find (𝑓 + 𝑔) (𝑥). Find(𝑓 ◦ 𝑔) (𝑥) and (𝑔 ◦ 𝑓) (𝑥). (𝑓 + 𝑔) (𝑥)= (3𝑥 2 − 4𝑥 + 5) + (2𝑥 3 + 6𝑥 − 2) (𝑓 ◦ 𝑔) (𝑥) = 𝑓[𝑔(𝑥)] = 2𝑥 3 + 3𝑥 2 + −4𝑥 + 6𝑥 + 5 − 2 Combine like terms = 4(𝑥 + 1)2 −5 𝑥 + 1 Replace all x’s in f(x) with x+1 = 2𝑥 3 + 3𝑥 2 + 2𝑥 + 3 Simplify. = 4 𝑥 2 + 2𝑥 + 1 − 5(𝑥 + 1) Expand the binomial = 4𝑥 2 + 8𝑥 + 4 − (5𝑥 + 5) Multiply Difference of Functions =4𝑥 2 + 8𝑥 − 5𝑥 + (4 − 5) Combine like terms = 4𝑥 2 + 3𝑥 − 1 Simplify Let 𝑓 𝑥 = 6𝑥 2 − 14𝑥 + 3 𝑎𝑛𝑑 𝑔 𝑥 = 2𝑥 2 + 6𝑥 − 2. Find (𝑓 − 𝑔) (𝑥). 𝑔 ◦ 𝑓 𝑥 = 𝑔[𝑓 𝑥 ] (𝑓 − 𝑔) (𝑥)= (6𝑥 2 − 14𝑥 + 3) − (2𝑥 2 + 6𝑥 − 2) = (4𝑥 2 −5𝑥) + 1 Replace all x’s in g(x) with 4𝑥 2 − 5𝑥 Note: Change the sign (additive inverse) of all terms = 4𝑥 2 − 5𝑥 + 1 Simplify of the subtrahend. = (6𝑥 2 − 2𝑥 2 ) + (−14𝑥 − 6𝑥) + (3 + 2) Combine like terms Note: (𝒇 ◦ 𝒈) (𝒙) and (𝒈 ◦ 𝒇) (𝒙)are two different things, but = 4𝑥 2 − 20𝑥 + 5 Simplify. some cases may happen that they are just equal.