General Mathematics 2nd Quarter PDF

GENERAL MATHEMATICS 2nd Quarter COURSE OUTLINE INVERSE FUNCTIONS EXPONENTIAL FUNCTIONS LOGARITHMIC FUNCTIONS SIMPLE INTEREST AND COMPOUND INTEREST  EXPLORING INVERSE FUNCTION GENERAL MATHEMATICS LEARNING OBJECTIVES AT THE END OF THE LESSON, I SHOULD BE ABLE TO: ØUNDERSTAND THE...

GENERAL MATHEMATICS 2nd Quarter COURSE OUTLINE INVERSE FUNCTIONS EXPONENTIAL FUNCTIONS LOGARITHMIC FUNCTIONS SIMPLE INTEREST AND COMPOUND INTEREST  EXPLORING INVERSE FUNCTION GENERAL MATHEMATICS LEARNING OBJECTIVES AT THE END OF THE LESSON, I SHOULD BE ABLE TO: ØUNDERSTAND THE CONCEPT OF INVERSE ØFIND THE INVERSE OF A FUNCTION ØVERIFY IF THE GIVEN FUNCTIONS ARE INVERSE OF EACH OTHER STEPS IN GETTING THE INVERSE OF A FUNCTION 1. REPLACE f(x) to y 2. INTERCHANGE x to y 3. SOLVE THE NEW VALUE OF y IN TERMS OF x 4. REPLACE y WITH −1 ( ) STEPS IN FINDING THE INVERSE OF A FUNCTION 1.) Change the name of the function f(x) to y 2.) Interchange the variables x and y 3.) If the remaining equation; i. doesn’t contain a fraction , then isolate the variable y on one side of the equation and the rest of the terms on the opposite side, then PROCEED TO STEP 5 ii. contains fraction, then multiply both sides by the denominator of the fraction, PROCEED TO STEP 4 4.) Simplify the left side of the equation by Direct Multiplication/ Distribution 5.) Check the remaining terms in the equation if there are terms with Y variable i. If none, then isolate the variable y to one side of the equation, and PROCEED TO STEP 7 ii. If there are terms with Y variable, then isolate the terms with y variable on one side of the equation and the rest on the opposite side 6.) Factor out the common variable Y in symbols y( ± ) 7.) Divide both sides so that the variable y will left isolated on one side of the equation 8.) Change the variable y to the symbol of inverse of the given function − (x) EXAMPLES: Find the inverse of each function 1. f(x) = 2x - 7 3 + 1 2. g(x) = 2 2 3. h(x) = −1 1+ 4. p(x) = 2 Guided Practice: Find the inverse of each function 1. f(x) = x + 6 −9 2. g(x) = 3 2+ 3. h(x) = 4. p(x) = −2 Guided Practice: Verify if the functions are inverse of each other 1. f(x) = x + 9 and −1 (x) = x - 9 3− 2. g(x) = and −1 (x) = 3 - 4x 4 −1 − + 4 3. h(x) = -2x + 4 and ℎ (x) = 2  EXPONENTIAL FUNCTIONS, EQUATIONS AND INEQUALITIES 2nd Quarter EXPONENTIAL FUNCTION Is a function involving exponential expression showing a relationship between the independent variable x and dependent variable y or f(x). An Exponential Function is defined by the rule f(x) = where a must not be equal to zero examples: ① f(x) = 2 + 3 ② y = 102 EXPONENTIAL EQUATION An Exponential Equation is an equation involving exponential expression that can be solved for all x values satisfying the equation. examples: ① 121 = 11 ② 3 = 9 − 2 EXPONENTIAL INEQUALITY Is an inequality involving exponential expression that can be solved for all x values satisfying the inequality. examples: ① 641 3 > 2 ② (0.9) > 0.81 Determine if EF, EE, EI − < + = 6 6>( ) + 1 < 10 y = 64 = + 100 > f(x) = g(x) = y = − ( ) + = ( ) f(x) = 7 = 27 < f(x) = + y = (-4) − = + + > 243 x = 8 = Solving exponential equation One-to-one Property of Exponential Functions states that in f(x) = , if 1 = 2 , then 1 = 2. ① 22 − 1 = 43 + 2 ② 43 + 1 = 8 − 1 − 1 ③ 16 = 64 Guided Practice: Solve the exponential equation. 1. 9 + 1 = 3 2. 33 = 32 − 4 3. 34 = 9 + 1 4. 25 + 1 = 125 5. 43 − 1 = 64 GRAPH OF EXPONENTIAL FUNCTIONS LEARNING OBJECTIVES At the end of the lesson, I should be able to; 1. Define and Identify the properties of exponential function 2. Determine the asymptotes of an exponential function 3. Construct a table of values of an exponential function 4. Graph exponential functions GRAPH OF EXPONENTIAL FUNCTIONS An EXPONENTIAL GRAPH is a curve that has a horizontal asymptote and it either has an increasing slope or a decreasing slope. PROPERTIES OF EXPONENTIAL FUNCTIONS 1. The DOMAIN is the set of all real numbers 2. It is a One-to-One relation. It satisfies the horizontal line test 3. The ASYMPTOTE of the graph is always horizontal 4. The function is increasing (Exponential Growth) if b > 1, and the function is decreasing (Exponential Decay) if 0 < b < 1 STEPS IN GRAPHING EXPONENTIAL FUNCTIONS 1st step: Determine the horizontal asymptote to the graph of the function 2nd step: Construct the table of values * Assign any values of x/Domains 3rd step: Substitute the values of x/Domains to the given function to find the values of y/Range 4th step: Plot points and Draw the Graph 5th step: Determine the graph if exponential Growth or Decay EXAMPLES Graph the given functions 1. f(x) = 2 + 4 2. g(x) = −2 + 3 1 3. h(x) = ( ) 2 INTRODUCTION TO LOGARITHMS 2nd Quarter INTRODUCTION TO LOGARITHMS GUIDE QUESTIONS: 1. What is logarithm? 2. How do we rewrite exponential expression in logarithmic form and vice versa? 3. How do we evaluate logarithms? LOGARITHM The logarithm to the base b is the inverse function of exponential with base b. That means the logarithm of a number Y to the base b is equal to a number X. logarithm is written in the form: = b = base X = exponent Y = argument note: The base must be a positive number not equal to 1. The argument must be a positive number  Rewriting Exponential Equations to Logarithmic Equations and Vice Versa The logarithmic form of y = is log = The exponential form of log = is y= Example: ① 25 = 32 ② log3 81 = 4 COMMON LOGARITHM The common logarithm is a logarithm with a base of 10. In this case, the base is not written anymore. Example: The exponential form of log 1000 = 3 3 is 10 = 1000 NATURAL LOGARITHM The natural logarithm is a logarithm with a base , In this case, the logarithm is written as Example: The exponential form of 2 a = 2 is = Rewrite the following logarithm in its exponential form and vice versa 1. 2 = 10 2. 102 = 100 3. log3 12 = x 4. log2 8 = 3 Evaluate the following logarithms 1 1. log2 256 6. log2 16 1 2. log3 243 7. log2 + log2 64 8 3. log 100 4. 3 log2 32 Laws of Logarithm GUIDE QUESTIONS: 1. What are the laws of logarithm? 2. How do we simplify logarithmic expressions using laws of logarithm? 3. How do we evaluate logarithmic expressions using laws of logarithm? Addition Law of Logarithm/ Logarithm of a Product The logarithm of a product is equal to the sum of the logarithms of its factors. log ( ) = log ( ) + log ( ) Example 1: Expand the logarithmic expression log 5 2: Express the logarithmic expression log3 9 + log3 3 into single term The subtraction Law of Logarithm/ Logarithm of a Quotient The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. log ( ) = log − log Example: 8 1. Expand the logarithmic expression log5 ( ) 2. Express the logarithmic expression (log5 + log5 ) − log5 3 as a single term The Exponent Law of Logarithm/ Logarithm of a Power The logarithm of a power is equal to the product of the exponent n and the logarithm of m to the given base b log ( ) = n log ( ) Example: 1. Expand the logarithmic expression (93 ) 1 2. Express the logarithmic expression [log8 + log8 ] as a single 3 term Express logarithmic expressions into a single term 1. log3 2 + log3 6 − log3 5 2. log3 − log3 3 3. 2 log + 3 log 4 Expand logarithmic expressions 1. log ( ) 2 3 2. log ( ) 4 5 Evaluating Logaritmic Expressions 1. Evaluate log3 162 − log3 2 4 2. Evaluate log3 27