Functions offer an interesting way to understand relationships between quantities. As a businessman, Mr. Cruz uses functions to predict the income of his store located in Balanga City. He used previous data to make models that could describe the relationship between number of customer (x) and the sales for each of his products. For product A, the income is represented by A(x)=420x-250, while the income for product B is given by B(x)=250x + 35.

Which of the following would be an example of a function?

If y=f(x), which of the following CANNOT be true?

Mr. Cruz bought a box of product D with a total cost of x pesos. If there are 13 pieces of product D inside the box, which would give the function for the price of each piece?

If in a certain day, Mr. Cruz was able to sell 20 pieces of product A, what was the sales for product A that day? 

Suppose you are working for Mr Cruz and a customer wants to buy 5 pieces of product B. How would you determine the total price of the purchase?

Mr. Cruz told you that the price of product D is the composition of the functions A(x) and B(x) wherein the function A is applied to the results of the function of B. How would this be expressed mathematically?

What is the value of (𝐵 ○ 𝐴)(2)?

For product C, Mr Cruz determines that the function C(x) can be found by multiplying the number of customers by P120. Which of the following represents the function C?

How much will be the total sales from product A and B in a certain period of time?

During the celebration of the Philippine Independence Day on June 12, 2021, the store had a record high in terms of both number of customers and sales. If on that day, the store received 200 customers, what was the total sales for Product A?

RATIONAL FUNCTIONS.

We can simply say that a rational function can be defined by a rational fraction wherein both numerator and denominator are polynomial functions. As an artist, Ms Dilao takes interest on the ratio of the sides of a rectangular canvas. She believes that this ratio could highlight the beauty of a painting. In one of her collections, she used different rectangular canvas following the function L(x)=5x+8 for the length and W(x)=4x-2.

Which represents the ratio of the length to the width of the rectangular canvas?

Suppose the ratio . 4𝑥−2 is equal to ½. What is x?

                                            5𝑥+8

In one of her exhibits, the ratio of the sides of the canvas is given by 𝑓(𝑥) = 2𝑥+1. What is the domain of the rational 𝑥−2 function?

What is the x- intercept of the graph of 𝑓(𝑥) = 2𝑥+1?

                                                                                                           𝑥−2

Which of the following is the y-intercept of the graph of the rational function 𝑓(𝑥) = 2𝑥+1?

                                                                                                                                                                                                 𝑥−2

Which of the following could be a solution if 𝑥+3/𝑥−2 > 0 ?

What is the solution set for the inequality 𝑥+3/𝑥−2 > 0?

The denominator of a fraction is 2 more than the numerator. If 1 is subtracted from both numerator and denominator, the resulting fraction has a value of 1/2. Find the original fraction.

ONE-TO-ONE AND INVERSE FUNCTIONS

Which of the following is a one-to-one function?

The function machine A follows the following instruction: add 3 to the input x and multiply the outcome to 4. If you are to make a function machine B which is the inverse of A, what will be its programming?

Which of the following functions is the inverse of 𝑓(𝑥) = 3/𝑥−2 given that 𝑓(𝑥) is one-to-one?

Marvin knows that to convert Celsius to Farenheit, he needs to multiply the given C by 9, divide the answer by 5 and then add 32. One time her mother called Marvin and told her that the temperature of his younger brother is 105.8 degrees Farenheit. Knowing only the formula to convert C to F and that the normal body temperature is 37 degree Celsius and below, should Marvin be worried about his brother?

EXPONENTIAL FUNCTIONS

The COVID-19 pandemic has changed many aspects of human life. To show how fast the virus spreads, Teacher Thelma makes a simple illustration, starting from 1 person infecting 2 others which in turn infect four more. She then explains how this rate can be explained using an exponential function.

Which of the following illustrates the number of new

Suppose 52x+1 > 125, what is x?

What is the asymptote of the graph of 𝑓(𝑥) = 3𝑥+1?

Teacher Thelma told her class about the element Francium which has a half-life of 22 minutes. If at a certain point a container has 100 grams of this element, how much of it will remain after 1 hour and 6 minutes?

The spread of the virus in a certain town can be predicted using the function 𝑓(𝑥) = 3𝑥+1 where f is the number of new cases and x is the number of days after transmission was initially detected. On what day can we start to expect the number of new cases to be over a thousand?

LOGARITHMIC FUNCTIONS

Earthquakes are frequent in the Philippines as it is situated in the Pacific Ring of Fire. The Moro Gulf earthquake in 1976 was the deadliest earthquake in the country recording a magnitude of 7.9 and 3,000 casualties. To measure the intensity of earthquakes, scientists use the Richter scale which is a logarithmic function. The magnitude (R) is determined by the logarithm of the ratio of the measure of the amplitude of the earthquake wave (A) and the amplitude of the smallest detectable wave (Ao).

Which of the following represents the function for the Richter scale?

Which of the following is the correct logarithmic form of 3−4 = 1/81?

Solve the inequality 𝑙𝑜𝑔3 (2𝑥 − 3) < 2.

What equation could represent the logarithmic function (1,0) (5, 1) (25,2) (125,3),…?

Given the logarithmic function D(m)= 𝑙𝑜𝑔3 (𝑚 + 5), which of the following CANNOT be a value of m?

What is the vertical asymptote of the logarithmic function 𝑓(𝑥) = 𝑙𝑜𝑔4 𝑥?

 If 𝑙𝑜𝑔 2 = 0.301 and 𝑙𝑜𝑔 3 = 0.477, find 𝑙𝑜𝑔 24.

Solve for x: 𝑙𝑜𝑔𝑏 𝑥 = 4 𝑙𝑜𝑔𝑏 2 + 𝑙𝑜𝑔𝑏 3

SIMPLE AND COMPOUND INTEREST

Wars do nothing but claim the life, peace, or happiness of all people involved. Countries exert effort to keep their economies stable. Businessmen will also try to keep their businesses open and keep up with the current market trend. For example, Mr de Guzman decides to take a loan of P5,000,000 to buy the raw materials for his factory.

Which of the following best illustrates a simple interest on Mr de Guzman's loan?

Suppose the loan of Mr de Guzman is given a simple interest rate of 10% per year by Bank A, how much will be the interest for the first year?

Suppose Bank B is offering Mr de Guzman 8% annual interest rate compounded semi-quarterly, while Bank C offers 6% interest rate compounded monthly. What is the interest earned by the money after 1 year if he chooses B?

If Mr de Guzman is to take the offer of Bank C, what will be the maturity value of his loan after two years? 

Which of the three banks should Mr de Guzman choose for his loan if he plans to pay it after a year?

SIMPLE AND GENERAL ANNUITIES

People are buying insurance either as a form of investment or to protect their investment. Mr. Reyes wants to make sure that when he grows old, he will still have a steady flow of income without having to work. Thus, he makes a series of payments, called annuities, to a financial institution in preparation for the future.

Which of the following illustrates a simple annuity?

If Mr Reyes pays ₱100.00 at the end of each month for 30 years on an account that pays interest at 10% compounded monthly, how much money will he have after 30 years?

Mr. Reyes received two offers for a lot he is selling. Company A offers payments of ₱50,000.00 downpayment and P1,000,000 lump sum payment after 5 years from now. Company B offered P50,000.00 downpayment plus 40,000 quarterly payments for 5 years. Compare the fair market value of the two offers if the money can earn 5% compouned annually. Which offer has a higher market value?

What is the period of deferral for a monthly payment of ₱75,000.00 for 4 years that will start 8 months from now?

STOCKS AND BONDS

The Philippine Stock Exchange has been very lately as a result of the war. Prices have become very unsteady which causes worries to stockholders. Mr Dimagiba owns a large share of Company X which is directly affected by the crisis. He owns 10,000 share of common stocks which he bought at P0.50 per share.

Which of the following shows what stocks are?

Which is true about the difference between stocks and bonds?

What was the dividend per share last year?

BUSINESS AND CONSUMER LOANS

People will rarely believe that the richest businessmen in the country or even in the world are borrowing money from financial institutions instead of using their own money. Business loans are common and it helps in ensuring the stability of the economy.

It is a method of paying a loan (principal and interest) on installment basis usually of equal amounts at regular interval.

Mr Sy borrowed ₱350,000.00 and he promised to pay in full after 3 years. If the interest rate is 7.5% compounded annually, how much should be paid after 3 years?

Suppose your loan for a car worth ₱899,000.00 requires a 20% down payment. How much is the mortgage?

Logical thinking is both an art and a science. We establish a community of reason though logic and it helps us in the organization of our thoughts and actions. Basically it is part of our daily lives.

Three friends are talking about their future dreams. The following are exerpts of their conversation. Which illustrates a proposition?

Given propositions

p: Mathematics is difficult.

q: Logic is Fun.

r: I love Math.

Which statement corresponds to ~ q?

Given propositions

p: Mathematics is difficult.

q: Logic is Fun.

r: I love Math.

Which statement corresponds to p ->~ r?

What symbolic form corresponds in the proposition “Either math is difficult or I love math”?

Given the propositions:

p: Elma is a teacher.

q: Arvin is a doctor.

If both propositions are true, what is the truth value of the proposition Elma is a teacher and Arvin is not a doctor?

Given the propositions:

p: Elma is a teacher.

q: Arvin is a doctor.

If proposition p is false while q is true, what is the truth value of the compound proposition Arvin is not a doctor or Elma is a teacher.

Which of the following is False if p is true and q is false?

Which of the following is the converse statement of the conditional proposition “If the test is easy, then Diane will pass the test”?

Which of the following is a tautology?