Chapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller

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Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller

1 Chapter 4 Page 1 of 16 Lecture Guide Math College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal Victor Valle College Last updated: 3/23/13

If a function is one-to-one, then it is (it has an inverse which is also a function).

1 #42 Determine whether the two functions are inverses. ( ) and ( ) 4.1 #22 4.

If ( ) is on ( ), then ( ) is on the graph of its inverse.

2 Chapter 4 Page 2 of Inverse Functions A function is if it passes both a vertical and horizontal line test. If a function is one-to-one, then it is (it has an inverse which is also a function). In eercises 22 and 24, determine if the relation defines as a one-to-one function of. 4.1 #42 Determine whether the two functions are inverses. ( ) and ( ) 4.1 # #24 To find the inverse of a function from its graph, reflect the graph across the line. If ( ) is on ( ), then ( ) is on the graph of its inverse. *Given the graph of ( ), graph its inverse #30 Use the definition of a one-to-one function to determine if ( ) is one-to-one

3 Chapter 4 Page 3 of 16 To find the inverse of a function from its equation, switch the and, and then solve for the "new". e. Write an equation for ( ). In eercises 52, a one-to-one function is given. Write an equation for the inverse function. 4.1 #52 ( ) f. Eplain wh the restriction is placed on. 4.1 #62 a. Graph ( ). g. Graph ( ) and ( ) on the same coordinate sstem. b. From the graph of, is a one-to-one function? c. Write the domain of in interval notation. h. Write the domain of in interval notation. d. Write the range of in interval notation. i. Write the range of in interval notation.

Chapter 4 Page 4 of 16 In eercises 56, a one-to-one function is given. Write an equation for the inverse function. 4.1 #56 ( ) 4.2 & 4.3 Eponential & Logarithmic Functions A.

4 Chapter 4 Page 4 of 16 In eercises 56, a one-to-one function is given. Write an equation for the inverse function. 4.1 #56 ( ) 4.2 & 4.3 Eponential & Logarithmic Functions A. Introduction is an eponential equation. is called the. To emphasize how eponential equations can increase rapidl, consider the following dream salar schedule in which a peron starts with a 2 salar on the first da, and ever da thereafter the salar is doubled. Avoiding Mistakes 4.1 #66 Given ( ), write an equation for. (Hint: Sketch ( ) and note the domain and range.) The base of an eponential function must not be negative to avoid situation where the function values are not real numbers. For eample, ( ) ( ) is not defined for because is not a real number. The base of an eponential function must not equal 1 because ( ) for all real numbers. This is a constant function, not an eponential function.

5 Chapter 4 Page 5 of 16 B. Graphing Eponential and Log Functions Graph Graph: The domain is:. The domain is:. The range is:. The range is:. Transformations of eponential functions: Convert each from logarithmic form to eponential form (or vice versa): Logarithmic Form Eponential Form 4.2 #26 Graph the function ( ) and write the domain and range in interval notation.

Chapter 4 Page 6 of 16 4.2 #38 Use the graph of to graph the function ( ). Write the domain and range in interval notation. b. Write the domain and range in interval notation. c.

6 Chapter 4 Page 6 of #38 Use the graph of to graph the function ( ). Write the domain and range in interval notation. b. Write the domain and range in interval notation. c. Determine the vertical asmptote. In eercises 86 and 92, write the domain in interval notation. 4.3 #86 ( ) ( ) Transformations of logarithmic functions: 4.3 #90 ( ) ( ) 4.3 #80 a. Use transformations of the graph of to graph ( ).

7 Chapter 4 Page 7 of 16 C. Evaluating Eponential and Log Functions 3. Eponential kes on calculator: Logarithmic kes on calculator: On our calculator, find the following values: 1. ( ) 4.3 #56 Estimate the value of each logarithm between two consecutive integers. Then use a calculator to approimate the value to 4 decimal places. For eample, is between 3 and 4 because a ( ) b. 4. c. In eercises 36-48, simplif the epression. 4.3 #36 d. 4.3 #42 ( ) e. ( ) 4.3 #48 ( ) f. ( ) Solve each equation: 1. Basic Properties of Logarithms 1. ( ) 2. ( ) ( ) 4.

8 Chapter 4 Page 8 of 16 In eercises 59-68, simplif the epression. 4.3 # #55 Suppose that $10,000 is invested with 4% interest for 5 r under the following compounding options. Complete the table. Compounding Option a. Annuall Value Result 4.3 #64 ( ) b. Quarterl c. Monthl d. Dail Compound interest: ( ) represents the. represents the. represents the. represents the. represents the. Use n= for interest compunded annuall. Use n= for interest compunded semi-annuall. Use n= for interest compunded quarterl. Use n= for interest compunded monthl. Use n= for interest compunded dail. e. Continuousl 4.2 #66 The population of Canada in 2010 was approimatel 34 million with an annual growth rate of 0.804%. At this rate, the population ( ) (in millions) can be approimated b ( ) ( ), where is the time in ears since a. Is the graph of an increasing or decreasing eponential function? b. Evaluate ( ) and interpret its meaning in the contet of the problem. c. Evaluate ( ) and interpret its meaning in the contet of the problem. Round the population value to the nearest million. d. Evaluate ( ), ( ), and ( ). For continuousl compounding interest, use the formaula

9 Chapter 4 Page 9 of Properties of Logarithms Properties of Logarithms: 1. ( ) product proper 2. ( ) quotient propert 3. ( ) power propert 4.4 #18 Use the product propert of logarithms to write the logarithm as a sum of logarithms. Then simplif if possible. 4.4 #34 Appl the power propert of logarithms. ( ) 4.4 #42 Write the logarithm as a sum or difference of logarithms. Simplif each term as much as possible. ( ) ( ) 4.4 #24 Use the quotient propert of logarithms to write the logarithm as a sum of logarithms. Then simplif if possible. 4.4 #62 Write the logarithmic epression as a single logarithm with coefficient 1, and simplif as much as possible. ( ) 4.4 #28 Use the quotient propert of logarithms to write the logarithm as a sum of logarithms. Then simplif if possible. ( ) 4.4 #62 Write the logarithmic epression as a single logarithm with coefficient 1, and simplif as much as possible. ( )

Chapter 4 Page 10 of 16 4.4 #74 Use,, and to approimate the value of the given logarithm. 4.5 Eponential and Logarithmic Equations 4.5 #16 Solve the equation. 4.5 #18 Solve the equation. ( ) 4.

10 Chapter 4 Page 10 of #74 Use,, and to approimate the value of the given logarithm. 4.5 Eponential and Logarithmic Equations 4.5 #16 Solve the equation. 4.5 #18 Solve the equation. ( ) 4.4 #80 a. Estimate the value of between two consecutive integers. For eample, is between 2 and 3 because. b. Use the change-of-base formula and a calculator to approimate the logarithm to 4 decimal places. For eercises 24-36, solve the equation. Write the solution set with the eact values given in terms of common or natural logarithms. Also give approimate solutions to 4 decimal places. 4.5 #24 c. Check the result b using the related eponential equation.

11 Chapter 4 Page 11 of 16 For eercises 24-36, solve the equation. Write the solution set with the eact values given in terms of common or natural logarithms. Also give approimate solutions to 4 decimal places. 4.5 #28 For eercises 42-60, solve the equation. Write the solution set with the eact values. Also give approimate solutions to 4 decimal places, if necessar. 4.5 #42 ( ) ( ) 4.5 #46 ( ) 4.5 #36 For eercises 42-60, solve the equation. Write the solution set with the eact values. Also give approimate solutions to 4 decimal places, if necessar. 4.5 #50 ( )

12 Chapter 4 Page 12 of #56 ( ) ( ) 4.5 #60 ( ) ( ) 4.5 #58 ( ) 4.5 #66 Use the model ( ). Barb puts aside $10,000 in an account with an interest reinvested monthl at 2.5%. How long will it take for her to earn $2000? Round to the nearest month.

13 Chapter 4 Page 13 of Modeling with Eponential and Logarithmic Functions 4.6 #14 Solve for the indicated variable. for (used in chemistr) 4.6 #24 Suppose that $50,000 from a retirement account is invested in a large cap stock fund. After 20 r, the value is $194, a. Use the model to determine the average rate of return under continuous compounding. 4.6 #18 Solve for the indicated variable. ( ) for (used in medicine) b. How long will it take the investment to reach one-quarter million dollars? Round to 1 decimal place.

14 Chapter 4 Page 14 of 16 Some Chapter 4 Review Problems 3. Graph: ( ) ( ) 1. Solve: ( ) ( ) 2. Graph: ( ) ( ) 4. Graph

15 Chapter 4 Page 15 of Solve: 7. Solve: ( ) 6. Solve: 8. Solve: ( ) ( )

16 Chapter 4 Page 16 of For continuousl compounding interest, at what interest rate should $500 be invested so that is grows to $750 in 8 ears? For f( ), find f 8 5 domain and range of f 1 ( ). 1 ( ), and the 10. For how long should $800 be invested at 4.3%, compounded dail, in order for it to grow to $2000?

Ch. 4 Review College Algebra Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 4 Review College Algebra Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Ch. Review College Algebra Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Decide whether or not the functions are inverses of each other. 3 5 +