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GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan BASIC CONCEPTS OF FUNCTIONS AND RELATIONS...

GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan BASIC CONCEPTS OF FUNCTIONS AND RELATIONS C.​ Vertical Line Test It used to check if the given graph of an equation represents a function. A.​ Relations -​ A graph represents a function IF A correspondence between two AND ONLY IF no vertical line sets and can be written in ordered intersects the graph at more than pairs. one point. The DOMAIN of a relation is the -​ This graph is a function because if set of first coordinates. It is also we have an imaginary vertical line, called “input” it will intersect the graph at only The RANGE of a relation is the set one point. of second coordinates. It is also called “output” REPRESENTATION OF REAL-LIFE SITUATION USING FUNCTION AS A MATHEMATICAL MODEL Example : R → {( 1, Pepsi), (2, Tropicana), (3, Coke)} A.​ Function as a Model 1.​ Visualize the Real – life situations and ​ Possible Types of Relations: determine the variables involved 2.​ Identify the relation between the variables and express its model into words 3.​ Set up the mathematical model 4.​ Use the mathematical model B.​ Some Types of Function B.​ Functions A relation in which each element of the domain corresponds to exactly one element of the range. -​ This is why ONLY One to One and Many to One are considered relations that are functions. One to Many is only a relation.  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan C.​ Sample Problem Fat and Kram are planning to prepare for a party in a ​ Special Types of Linear Function: place near SRCCMSTHS. It costs them 12,000 PHP for Constant Function - a linear function f is a the venue and an additional 200 PHP per guest. State constant function if f(x) = mx + b, where m = the cost C as a function of x number of guests and 0 and b is any real number. Thus f(x) = b. determine the total expenses of Fat and Kram if Real-Life Example: A fixed amount of they have 50 guests in the party. tricycle fare F in any distance x within your barangay. Note: Use the Function as a Model steps above to have a guide on solving problems. -​ Make C as a function of x, this will be the overall cost or the total expenses. -​ And in order to get the cost, in real life, we multiply the price with the number of attendees or guests. -​ We should also take note of the additional payment like the P12,000 for the venue to Identity Function - a linear function f is an add to the overall cost. identity function if f(x) = mx + b, where m=1 -​ In this way, we can have: and b = 0. Thus f(x) = x. 𝐶(x) = 200x + 12,000 To solve the problem: -​ Make x equal to 50 as the number of guests. 𝐶(50) = 200(50) + 12,000 = P22,000 D.​ Linear Function A function in the form f(x) = mx + b, where m and b are real numbers, and m and f(x) are not both equal to zero. The example above is a linear E.​ Sample Problem function. Sample Problem: 𝑓(𝑥) = 𝑚𝑥 + 𝑏 A Volleyball Sports Complex has seats for Where: 10,000 people. It is filled to capacity for each game and the ticket costs 200 PHP. The 𝑦2−𝑦1 Slope = 𝑚 = 𝑥2−𝑥1 owner wants to increase the ticket price. He predicts that if the ticket price is increased by 20 PHP, the number of people who will  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan see the game will decrease by 250. Write a mathematical model to describe the F.​ Piecewise Function owner’s income after an increase in his A function defined by multiple ticket price and determine the greatest sub-functions, where each sub-function applies to a possible income that can be received by the certain interval of the main function’s domain. owner. -​ Make I as a function of x, this is the owner’s income. -​ Take note that: If the P200 price is increased by P20, the expected Sample Problem: 10,000 people will then decrease A resort offers an entrance fee of P300 per by 250. head if there are fewer than 15 persons -​ With this information, going inside the place. However, if there are we can make: 15 persons or more, one of them will be free 𝐼(𝑥) = (200 + 20𝑥)(10, 000 − 250𝑥) of charge. Write a piecewise-defined model of the total amount paid A as a function of Use foil method to get: the number of people x who entered the 2 resort. How much will be paid if there are 𝐼(𝑥) = − 5000𝑥 + 150000𝑥 + 2000000 20 people in the resort? In order to determine the greatest or maximum -​ Make A as a function of x, this is possible income of the owner, we use the formula of the total amount paid. Vertex (This is because we know that the graph of a -​ Take note of the conditions with quadratic equation is a parabola and if you can the pricing like if it’s less than 15 remember, the highest point of a parabola facing people attending, the entrance fee downwards is its vertex) is P300 per head. If it’s 15 or more, −𝑏 −150000 𝑣= 2𝑎 = 2(−5000) = 15 one will be free so we subtract 300 from the function. [The answer 15 is the maximum times that the owner can increase the price, increasing above 15 will give him a less income] Since 20 is Substitute x with 15 in the function above to have greater than 15, we should use the second one. your final answer. Substitute x with 20 with the function. 2 𝐼(15) = − 5000(15) + 15000(15) + 2000000 𝐴(20) = 300(20) − 300 𝐴(20) = 𝑃5, 700 𝐼(15) = 𝑃3, 125, 000 The amount to be paid is P5,700 The owner’s maximum income is P3,125,000  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 2 G.​ GENERAL TIPS 2. Let 𝑔(𝑥) = − 𝑥 − 𝑥 + 1. 1.​Read the problem carefully to gain a full Find each function value: understanding of the given situation. It is a.​ g (−1) helpful to construct an illustration of the b.​ g(w) situation or you can also create your own c.​ g (x + 1) specific example of a situation that involves a Simply substitute the given value to x in the function. similar scenario to the given. 2.​Use proper representations for the quantities a.​ g(-1) 2 involved in the given situation. Clearly 𝑔(𝑥) = −𝑥 − 𝑥 + 1 determine the independent variable to be 2 𝑔(− 1) = − (− 1) − (− 1) + 1 represented as x and the function you need to 𝑔(− 1) = 1 obtain which is symbolized by f(x). 3.​Take note of all the numerical facts known about the variable and the function value. 3. Consider the piecewise function 4.​From the information in step 3, form an , evaluate the function at: equation that defines the relationship a.​ x = −2 between the two quantities. b.​ x = 0 5.​Use the mathematical model to solve the c.​ x = 5 problem and write a conclusion that answers Simply find a sub-function that has a condition that the questions of the problem. aligns with the value of x. a.​ x = -2 Since -2 is less than 0, we EVALUATION OF FUNCTION should use the first function and substitute x with 2. -​ The process of replacing the variable in the 2 𝑓(𝑥) = 𝑥 + 1 function with a value from the function's 2 domain and computing the result. 𝑓(− 2) = (− 2) + 1 -​ Simply, it is the process of finding the value 𝑓(− 2) = 5 of f(x) or y that corresponds to a given value of x. OPERATIONS OF FUNCTION A.​ Example 1. Consider the given function y = f(x) defined by the set of ordered pairs. Evaluate the function at the given values of x: {(−3, 14) , (−1, 6) , (0, 5) , (2, 9) , (4, 8) , (7, 9)} a.​ x = 2 b.​ x = −1 c.​ x = 0 To solve, simply look at the ordered pairs and find the corresponding y value of the given x value.  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan (𝑥+1)(𝑥+4) (𝑓 ÷ 𝑔)(𝑥) = (𝑥+1) (𝑓 ÷ 𝑔)(𝑥) = 𝑥 + 4 Few Tips -​ For addition and subtraction, consider like terms and perform the rules of Addition or OPERATIONS OF FUNCTION: Composition & Decomposition Subtraction of polynomials A.​ Composition -​ For multiplication and division, apply the The composition of the function f with g is denoted rules of Multiplication and Division of by f ∘ g and is defined by the equation: Polynomials. Different ways of Factoring and (𝑓 ◦ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) Special Products can be applied. -​ The symbol f ∘ g is read as ‘‘f circle g’’ -​ In computing for f ∘ g , simply replace x in f A.​ Example with g(x) 2 Given: 𝑓(𝑥) = 𝑥 + 5𝑥 + 4 Example 𝑔(𝑥) = 𝑥 + 1 Givens: 𝑓(𝑥) = 1 − 𝑥, ℎ(𝑥) = 𝑥 − 2𝑥 + 1 2 Determine: What is (ℎ ◦ 𝑓)(𝑥) ? 1.​ (𝑓 + 𝑔)(𝑥) 2 2 (ℎ ◦ 𝑓)(𝑥) = ℎ(𝑓(𝑥)) = (1 − 𝑥) − 2(1 − 𝑥) + 1 (𝑓 + 𝑔)(𝑥) = (𝑥 + 5𝑥 + 4) + (𝑥 + 1) 2 2 (ℎ ◦ 𝑓)(𝑥) = ℎ(𝑓(𝑥)) = 𝑥 − 2𝑥 + 1 − 2 + 2𝑥 + 1 (𝑓 + 𝑔)(𝑥) = 𝑥 + 5𝑥 + 𝑥 + 4 + 1 2 2 (ℎ ◦ 𝑓)(𝑥) = ℎ(𝑓(𝑥)) = 𝑥 (𝑓 + 𝑔)(𝑥) = 𝑥 + 6𝑥 + 5 B.​ Decomposition 2.​ (𝑓 − 𝑔)(𝑥) A process of breaking down one complex function 2 (𝑓 − 𝑔)(𝑥) = (𝑥 + 5𝑥 + 4) − (𝑥 + 1) into multiple smaller functions. 2 (𝑓 − 𝑔)(𝑥) = 𝑥 + 5𝑥 − 𝑥 + 4 − 1 Example 2 2 (𝑓 − 𝑔)(𝑥) = 𝑥 + 4𝑥 + 3 1.​ Given: 𝑓(𝑥) = 𝑥−3 2 We can make ℎ(𝑥) = 𝑥+5 as our outside function 3.​ (𝑓 × 𝑔)(𝑥) 2 ℎ(𝑥) so that we can have 𝑔(𝑥) = 𝑥 − 2 as our (𝑓 × 𝑔)(𝑥) = (𝑥 + 5𝑥 + 4)(𝑥 + 1) 3 2 2 inside function. In this way, we see that: (𝑓 × 𝑔)(𝑥) = 𝑥 + 𝑥 + 5𝑥 + 5𝑥 + 4𝑥 + 4 2 2 3 2 ℎ(𝑔(𝑥)) = (𝑥−2)+5 = 𝑥−3 = 𝑓(𝑥) (𝑓 × 𝑔)(𝑥) = 𝑥 + 6𝑥 + 9𝑥 + 4 2 2.​ Given: 𝑓(𝑥) = 𝑥 + 6𝑥 + 8 4.​ (𝑓 ÷ 𝑔)(𝑥) Find a function that is close with the given but is a 2 perfect square trinomial. For this problem, I chose (𝑓 ÷ 𝑔)(𝑥) = (𝑥 + 5𝑥 + 4) ÷ (𝑥 + 1) 2 2 2 Factor (𝑥 + 5𝑥 + 4) = (𝑥 + 1)(𝑥 + 4) (𝑥 + 3) = 𝑥 + 6𝑥 + 9. Take the 𝑥 + 3 as your inside function ℎ(𝑥). We can also observe that the 2 given and 𝑥 + 6𝑥 + 9 has a difference of − 1,  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 2 because of this, we can make 𝑥 − 1 as our outside D.​ Rational Inequality function 𝑔(𝑥). A rational inequality is a mathematical statement that is composed of rational expressions With this way, we can have: combined with a < , > , sign. 2 2 𝑥−1 1 𝑔(ℎ(𝑥)) = (𝑥 + 3) − 1 = (𝑥 + 6𝑥 + 9) − 1 Example: > 2 𝑥+7 2 𝑔(ℎ(𝑥)) = 𝑥 + 6𝑥 + 8 = 𝑓(𝑥) E.​ Rational Function A rational function can be written in the RATIONAL FUNCTION 𝑛(𝑥) form of 𝑓(𝑥) = 𝑑(𝑥) , where n x and d(x) are A.​ Polynomial polynomials where d(x) ≠ 0 A polynomial with a degree n is an algebraic Tips in obtaining the Domain of a expression that can be written in the form: Rational Function: 𝑛(𝑥) The domain of a rational function 𝑓(𝑥) = 𝑑(𝑥) is all the values of x that will not make d(x) equal to zero. So to obtain the domain of a rational function: 1.​ Set the denominator equal to zero and solve for x. 2.​ Set the domain as all real numbers excluding the value/s of x obtained in step B.​ Rational Expression A rational expression is an algebraic 𝑎 DOMAIN AND RANGE OF RATIONAL FUNCTION expression that can be written in the form 𝑏 where a and b are polynomials & b ≠ 0. It must always be simplified to its lowest term. -​ The DOMAIN of a rational function 𝑛(𝑥) 𝑥 𝑓(𝑥) = 𝑑(𝑥) is all the values of x that will Example: 𝑥+8 not make 𝑑(𝑥) equal to zero. C.​ Rational Equation -​ The RANGE of a rational function can be A rational equation is an equation that found by first finding the inverse of the contains one or more rational expressions. These function and determining its domain. If this rational expression/s may be on one or both sides of does not work, the best way is to graph the the equation. Extraneous solution is an apparent rational function. solution that does not solve its equation. 𝑥−1 1 A.​ Domain Example: 𝑥+7 = 2 Steps to find the Domain: 1.​ Set the denominator equal to zero and solve for x.  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 2.​ Set the domain as all real numbers excluding 𝑥+8 0= 𝑥−2 the value/s of x obtained in step 1. 0=𝑥+8 Example: 3 𝑥= −8 𝑓(𝑥) = 𝑥+8 The graph of the function crosses the x-axis at (-8,0) 𝑥+8=0 𝑥= −8 Important Reminders: Domain: 𝑓 = {𝑥ϵ𝑅/𝑥 ≠− 8} ❑It is possible that a graph of a rational function does not have any intercept. This simply means that B.​ Intercepts the graph will not cross any axis of the cartesian Intercepts are x or y coordinates of the plane. points at which a graph crosses the x-axis or y-axis, ❑ It is important to check whether a computed respectively. x-intercept is a part of the domain of the function. If ❑ y-intercepts: y-coordinate of the point where the the computed value for x-intercept is not part of the graph crosses the y-axis domain, then do not use the value as an intercept. ❑ x-intercept: x-coordinate of the point where the graph crosses the x-axis C.​ Holes A hope on a graph looks like a hollow circle. Steps to find the Intercepts of a It represents the fact that the function approaches Rational Function: the point but is not actually defined on that precise x value. 1.​ To find the y-intercept, substitute 0 for x -​ If the numerator and denominator and solve for y or f(x) have similar factor, there exists a 2.​ To find the x-intercept , substitute 0 for y hole in the graph of function. and solve for x. Steps to find the Holes of a Example: Rational Function, if any: 𝑥+8 1.​ To find the x-coordinate of the hole, set the 𝑓(𝑥) = 𝑥−2 “common factor” to zero and solve for x. a.​ y-intercept 2.​ To find the y-coordinate of the hole, simplify Substitute 𝑥 with 0: the given rational function by crossing out (𝑜)+8 8 the “common factor”. Then, substitute the 𝑓(𝑥) = (𝑜)−2 = −2 computed x- coordinate into the simplified 𝑓(𝑥) = − 4 rational function and solve for f(x) or y. The graph of the function crosses the y-axis at (0,-4) Example: (3𝑥+1)(𝑥−1) b.​ x-intercept 𝑓(𝑥) = (𝑥−1) Substitute 𝑦 or 𝑓(𝑥) with 0:  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan a.​ x coordinate Example: To get this, simply find the common factor/s 2 𝑥 +6𝑥+8 and equate it to 0. For this, 𝑥 − 1 is our common 𝑓(𝑥) = 2 𝑥 −𝑥−6 factor. a.​ Step 1 𝑥−1=0 Factor both numerator and denominator: 𝑥=1 2 𝑥 +6𝑥+8 (𝑥+4)(𝑥+2) 𝑓(𝑥) = 2 = (𝑥−3)(𝑥+2) 𝑥 −𝑥−6 b.​ y coordinate Remove the common factor from the b.​ Step 2 function and solve the 𝑓(𝑥) 𝑜𝑟 𝑦 using what’s Identify the restrictions. These are the values that remaining. Substitute 𝑥 with the value we got above make the denominator equal to zero. which is 1. (𝑥+4)(𝑥+2) 𝑓(𝑥) = (3𝑥+1)(𝑥−1) 𝑓(𝑥) = (𝑥−3)(𝑥+2) (𝑥−1) 𝑓(𝑥) = 3𝑥 + 1 𝑥−3=0 𝑥+2=0 𝑦 = 3(1) + 1 = 4 𝑥≠3 𝑥 ≠− 2 The coordinates of the hole is (1,4) c.​ Step 3 Equate the numerator to zero and solve for x to D.​ Zeroes identify the values of the independent variable x The value of x that will make the numerator that make the numerator equal to zero. zero without simultaneously making the 𝑥+4=0 𝑥+2=0 denominator equal to zero. 𝑥 =− 4 𝑥 =− 2 d.​ Step 4 Steps to find the Zeroes of a The zeroes of the equation are results from Step 3. Rational Function: BUT -2 is not included because it’s also a restriction 1.​ Factor the numerator and denominator of for x. the rational function if possible The zero of the rational function is only − 4 2.​ Identify the restrictions of the rational [When - 4 is substituted to the independent variable functions. x of the given function, the corresponding value of 3.​ Identify the values of the independent f(x) is zero.] variable x that make the numerator equal to E.​ Asymptotes zero.(Simply equate your numerator to An asymptote is an imaginary line to which zero) a graph gets closer and closer as it increases or 4.​ The zeroes of the rational function are the decreases its value without limit. values of x that make the numerator zero ❑ Vertical Asymptote: The line x = a is a but are not restrictions of the rational vertical asymptote if the graph increases or function. decreases without bound on one or both sides of the line as x moves in closer and closer to x = a. It is the  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan restriction/s on the x values of a reduced rational function. ❑ Horizontal Asymptote: The line y = b is a horizontal asymptote if the graph approaches y = b Rules for obtaining Horizontal Asymptotes as x increases or decreases without bound. Note that of a Rational Function: it doesn’t have to approach y = b as x BOTH increases -​ To find the horizontal asymptote of a and decreases. It only needs to approach it on one rational function, compare the degree n of side in order for it to be a horizontal asymptote. the numerator n(x) with the degree m of the denominator d(x) and then, consider Steps to find the Vertical Asymptote the following rules: of a Rational Function: 1.​ Obtain the factors of the numerator and the ❑ If n < m, the horizontal asymptote denominator. we get is y=0. 2.​ Simplify the form of the rational function by ❑ If n = m, the horizontal asymptote is cancelling out the common factor/s the ratio of the leading coefficient of the between the numerator and denominator. numerator 𝑎𝑛to the leading coefficient of the -​ Note: The factors that are canceled when a rational function is reduced denominator 𝑏𝑚 represent holes in the graph. ❑ If n > m, the graph has no horizontal 3.​ To find the vertical asymptote, equate the asymptote. denominator of the reduced rational function to 0 and solve for x Example: 4𝑥 Example: 𝑓(𝑥) = 2𝑥+1 Since 𝑛 = 𝑚, 2 𝑥 −4𝑥+3 𝑎𝑛 𝑓(𝑥) = 4 2 𝑥 +3𝑥+2 HA = 𝑏𝑚 = 2 =2 𝑦=2 a.​ Step 1 Get the Factor: 2 𝑥 −4𝑥+3 (𝑥−1)(𝑥−3) 𝑓(𝑥) = 2 = (𝑥+1)(𝑥+2) 𝑥 +3𝑥+2 Rules for obtaining Oblique Asymptotes b.​ Step 2 of a Rational Function, if any: Cancel the common factor if there is. There is none Exists when the numerator of f(x) has a in this example. degree that is one higher than the degree of the c.​ Step 3 denominator. To find its equation, consider the Equate the denominator to 0 and solve for x following steps: 𝑥+1=0 𝑥+2=0 1.​ Divide the numerator by the denominator. 𝑥 =− 1 𝑥 =− 2 These are the vertical asymptotes.  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 2.​ The oblique asymptote is y equal to the 5.​ Sketch the graph: use the plotted points quotient from step 1 with the remainder and the asymptotes as guides. ignored. Example: 2 𝑥 −4 Example: 𝑓(𝑥) = 2 Find the following: −9+𝑥 2 𝑥 +8𝑥+7 ℎ(𝑥) = 𝑥+2 Dom x-inter y-inter Zeroes All real (+2, 0) & 4 +2 & -2 (0, 9 ) nos. except ±3 (-2,0) a.​ Step 1 VA HA OA Holes Divide the numerator with the denominator x = 3 & x = -3 Y=1 none none Follow the guidelines above and use these informations as your guide in plotting b.​ Step 2 The oblique asymptote is the quotient with remainder ignored and set to y. That is, 𝑦 = 𝑥 + 6 GRAPH OF RATIONAL FUNCTIONS INVERSE OF A FUNCTION Guidelines in Graphing A.​ Inverse Rational Functions: A relation reversing the process performed 1.​ Determine the vertical, horizontal, and −1 by any function 𝑓(𝑥) is called inverse 𝑓 oblique asymptotes if there are any. 2.​ Locate the holes if there are any 3.​ Find the intercepts and plot them. B.​ How to get the Inverse 4.​ Plot points to the left, to the right, and 1.​ Replace 𝑓(𝑥) with y between the vertical asymptotes. 2.​ Interchange x and y  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 3.​ Solve for new y from step 2 −1 4.​ Replace the new y with 𝑓 (𝑥) if the inverse is a function. Example: 𝑥+8 𝑓(𝑥) = 2𝑥+3 Simply follow the steps above :) 𝑥+8 𝑦= 2𝑥+3 𝑥(2𝑦 + 3) = 𝑦 + 8 𝑦+8 𝑥= 2𝑦+3 2𝑥𝑦 + 3𝑥 = 𝑦 + 8 2𝑥𝑦 − 4 = 8 − 3𝑥 8−3𝑥 𝑦(2𝑥−1) 8−3𝑥 𝑦= 2𝑥−1 2𝑥−1 = 2𝑥−1 −1 8−3𝑥 𝑓 (𝑥) = 2𝑥−1 EXPONENTIAL EQUATIONS AND INEQUALITIES C.​ Properties of Inverse Function −1 1.​ 𝑓 (𝑥) is a one-to-one function; 𝑓(𝑥) is also A.​ Exponential Equations a one-to-one function. −1 An equation in which a variable occurs in an 2.​ The domain of 𝑓 (𝑥) = Range of 𝑓(𝑥) & exponent. The key in solving exponential equations is −1 the range of 𝑓 (𝑥) = Domain of f (x) to write both sides of the equation as powers of the 3.​ f and g are inverses of each other if: same base. 𝑓(𝑔(𝑥)) = 𝑥; for every x in the domain of Example: g 𝑥 𝑥−2 𝑔(𝑓(𝑥)) = 𝑥; for every x in the domain of f 3 = 27 4.​ The graph of the inverse is the reflection of Write both sides as powers of the same base: 𝑥 3(𝑥−2) 3 the graph of the original function. The axis 3 =3 (Note: 3 = 27) of symmetry is the line y = x. 𝑥 3(𝑥−2) 3 =3 LAWS OF EXPONENTS Solve for x (from the exponents): 2𝑥 6 𝑥 = 3𝑥 − 6 2 = 2 3𝑥 − 𝑥 = 6 𝑥=3 Property of Equality: Let a, b and c be real numbers 𝑏 𝑐 and a ≠ 0. Then, 𝑎 = 𝑎 if and only if b = c.  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan B.​ Exponential Inequality An inequality in which a variable occurs in an exponent. The key in solving exponential EXPONENTIAL FUNCTION IN THE REAL WORLD inequality is to write both sides of the equation as powers of the same base. 𝑥 Example: A.​ Exponential Model: 𝑦 = 𝑎𝑏 2𝑥+7 2𝑥−3 y → is the total amount 4 ≤ 32 a → is the initial amount Write both sides as powers of the same base: 𝑛𝑒𝑤 𝑣𝑎𝑙𝑢𝑒 b → is the 𝑔𝑟𝑜𝑤𝑡ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 2(2𝑥+7) 5(2𝑥−3) 2 ≤2 x → is number of times of increasing/decreasing the Solve for x: amount 2(2𝑥 + 7) ≤ 5(2𝑥 − 3) 4𝑥 + 14 ≤ 10𝑥 − 15 Sample Problem: 6𝑥 29 29 Emerson deposits P50,000 in a savings 6 ≥ 6 𝑥≥ 6 account. The account pays 6% annual EXPONENTIAL FUNCTION interest. 𝑥 A. Represent the account balance of A function that can be written as 𝑓(𝑥) = 𝑏 Emerson where b> 0, b≠1, and x is any real number. B. If he makes no more deposits and no b → constant or base withdrawals, calculate his new balance after x → an independent variable called the exponent 10 years. Example: 𝑥 𝑓(𝑥) = 3 and 𝑓(𝑥) = 2𝑥 + 1 Use the formula and represent just like our a is P50,000 because it’s our initial amount. Also TRANSFORMATION OF GRAPHS OF A FUNCTION calculate the growth factor. 𝑥 The movement of graph from one location A.​ 𝑦 = 50, 000 × (1. 06) to another location To calculate his balance after 10 years, substitute x with 10 because it’s the number of times increased. 10 B.​ 𝑦 = 50, 000 × (1. 06) 𝑦 = 𝑃89, 542. 38 We can also use the formula: 𝑟 𝑘𝑡 𝑓(𝑡) = 𝑃(1 + 𝑘 ) Note: K represents the number of times per year that Interest is compounded  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan 5.​ 𝑙𝑜𝑔𝑏𝑀𝑁 = 𝑙𝑜𝑔𝑏𝑀 + 𝑙𝑜𝑔𝑏𝑁 𝑀 Other Common Example of Exponential Model 6.​ 𝑙𝑜𝑔𝑏 𝑁 = 𝑙𝑜𝑔𝑏𝑀 − 𝑙𝑜𝑔𝑏𝑁 𝑝 7.​ 𝑙𝑜𝑔𝑏𝑀 = 𝑝𝑙𝑜𝑔𝑏𝑀 𝑙𝑜𝑔𝑐𝑎 8.​ 𝑙𝑜𝑔𝑏𝑎 = 𝑙𝑜𝑔𝑐𝑏 B.​ Logarithmic Equation An equation that contains Logarithmic Expression. Some tips in solving logarithmic equation: 1.​ Consider Property of equality for logarithmic equations: Let m, n and b be real numbers where b can be any positive real number except 1, 𝑙𝑜𝑔𝑏𝑚 = 𝑙𝑜𝑔𝑏𝑛, if LOGARITHMIC FUNCTION and only if m = n 2.​ If possible, simply convert the logarithmic An exponent which b must have to produce a 𝑐 equation into an exponential equation and 𝑙𝑜𝑔𝑏𝑎 = 𝑐 if and only if bc 𝑏 = 𝑎 for b≠ 1 & b > 0 solve for the unknown. -​ In both the logarithmic and exponential Example: forms, b is the base. 2 -​ In exponential form, c is the exponent. 𝑙𝑜𝑔(25 + 𝑥 ) − 2𝑙𝑜𝑔(4 − 𝑥) = 0 -​ In 𝑙𝑜𝑔𝑏𝑎, a cannot be negative. -​ The value of 𝑙𝑜𝑔𝑏𝑎 can be negative. C.​ Logarithmic Inequality An inequality that contains Logarithmic Expression. Some tips in solving logarithmic equation: 1.​ Consider the ff. ideas: For b > 1 : A.​ Laws of Logarithm: m < n if 𝑙𝑜𝑔𝑏𝑚 < 𝑙𝑜𝑔𝑏𝑛 1.​ 𝑙𝑜𝑔𝑏1 = 0 For 0 < b < 1 : 2.​ 𝑙𝑜𝑔𝑏𝑏 = 1 m < n if 𝑙𝑜𝑔𝑏𝑚 > 𝑙𝑜𝑔𝑏𝑛 𝑥 2.​ Ensure that the logarithms are defined. 3.​ 𝑙𝑜𝑔𝑏𝑏 = 𝑥 3.​ Ensure that the inequality is satisfied. 𝑥 𝑙𝑜𝑔𝑏 4.​ 𝑏 =𝑥  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan Example: The magnitude R of an earthquake is given by 2 𝐸 𝑙𝑜𝑔3(3𝑥 − 5) > 𝑙𝑜𝑔3(𝑥 + 5) 𝑅= 𝑙𝑜𝑔 4.40 , where E in joules is the 3 10 energy released by the earthquake. D.​ Logarithmic Function Question: Suppose that an earthquake released 12 A function that can be written as approximately 10 joules of energy. What is the 𝑓(𝑥) = 𝑙𝑜𝑔𝑏𝑥 , where b > 0 and b ≠ 1. Logarithmic magnitude on a Richter scale? function is the inverse of exponential function. Use the given: 12 2 10 𝑅= 3 𝑙𝑜𝑔 4.40 SOLVING LOGARITHMIC EQUATIONS 10 In solving logarithmic function, follow the 𝑅 = 5. 06 tips in solving mentioned before. Example: 𝑙𝑜𝑔𝑥(2401) = 4 4 Others: 𝑥 = 2401 For sound intensity, use 4 4 4 In acoustics, the decibel (dB) level of a sound is: 𝑥 = 2401 𝑥=7 1 𝐷 = 10𝑙𝑜𝑔 −12 10 Basic Properties of 𝑓(𝑥) = 𝑙𝑜𝑔𝑏𝑥 For pH level measures, use: In chemistry, the pH level measures the acidity of a water-based solution that is measured by + the concentration of hydrogen ions [𝐻 ] in the solution: + 𝑝𝐻 = − 𝑙𝑜𝑔[𝐻 ] END :) LOGARITHMIC FUNCTION IN THE REAL WORLD Example: Earthquake magnitude on a Richter scale  GENERAL WEEK 1-8 MATHEMATICS 2ND SEMESTER | Ma’am Herwina G. Limpasan