Intermediate Algebra Chapter 12 Review

Transcription

1 Intermediate Algebra Chapter 1 Review Set up a Table of Coordinates and graph the given functions. Find the y-intercept. Label at least three points on the graph. Your graph must have the correct shape. For each exponential function, give the equation of the horizontal asymptote. 1. f x. g x ( ) = x ( ) = f(x) = 1+ x 4. f( x) = log x 5. g( x) = log 1 x 6. g( x) = 3 x+ x 7. Given f(x) = x and g(x) = x 9, find each of the following: a. f(9) + g( 1) b. f(x) + g(x) c. f(5) g(x) d. (f o g)(5) e. (g o f)(16) 8. Find the equation of the inverse of the given one-to-one functions: a. f(x) = x + 5 b. f(x) = (x 4) 3 3 c. f(x) = x + 1

2 9. Write each of the following as an exponential equation. a. log 3 = 1 5 b. log 3 81= 4 c. log = 10. Write each of the following as a logarithmic equation. a. 3 4 = 1 81 b. 5 1/ = Graph the inverse of the given one-to-one function on the same set of axes. Label at least three points on the graph. 1. What test do you use to determine if a given graph represents a function that has an inverse function? State the test.

3 13. Determine whether each graph represents a one-to one function (If so, it the function has an inverse function). State your conclusion. a. b. c.

4 Find the value of each of the following without using a calculator. Leave your answers in exact form (no logarithms in any answer). 14. log log log log log log log lne. log log e ln log 3 log 8x ( ) Solve each equation algebraically without using a calculator. Give exact answers. 7. log 4 x = 3 8. log x 16 = 4 9. log 9 3 = x 30. log = x 31. State the product rule, quotient rule, and power rule for logarithms. (See section 5, chapter 1.) Write each expression as a sum or difference of logarithms. Don t use a calculator; but, where possible, evaluate logarithm expressions. (Example: If the answer contains log 5 5, simplify to obtain ). 3. log 3 x 33. ln 5 e

5 34. log 5 5 y log b x z log 3 x x 1 ( ) Write as a single logarithm. Don t use a calculator; but, where possible, evaluate logarithm expressions without a calculator log b x + log b y lnx + lny log b ( x + 1) 3log b y 40. 3log x + 1 log ( y + 1 ) log y Solve each equation algebraically. Do not use a calculator, and give all answers in exact form (no logarithms allowed) x = x = x = x = x+1 = 36 Use a calculator to approximate the following to four decimal places. (Hint: The change-of-base formula may be needed on a few of these.) 46. e log log log ln4.8

6 Solve each equation by using an appropriate method. After you have gotten an exact answer, use your calculator to approximate the answer to the nearest thousandth (three decimal places) x = x = e x = x = x 1 = x+1 = 5 x 57. log ( 3x + 1) = log(x 3) = lnx = log 5 x log 5 ( 4x 1) = log 4 ( x + ) log 4 ( x 1) = 1 6. ln x + 4 = 1 Solve each of the following application problems algebraically. For each problem, you may use your calculator but must show enough work to outline your solution strategy. 63. How much will an investment of $15,000 be worth in 30 years if the annual interest rate is 5.5% and compounding is a. quarterly? b. monthly? c. continuously? 64. The equation y = (1.01) x models the population of the United States from 1950 through 003. In this equation, y is the population in millions and x represents the number of years after Use the equation to estimate the population of the United States in (Use a calculator and round your answer to two decimal places).

7 Answers: 1-6. See graphs on following pages. 7a. 5 7b. x + x c. 5x 45 7d e. 7 8a. f 1 (x) = 5x b. f 1 3 (x) = x c. f 1 (x) = x3 1 9a. 3 1/ 5 = b. 3 4 = c. 5 = a. log 3 81 = b. log 5 5 = See graph on following pages. 1. Use the horizontal line test HLT: A function f is one-to-one and thus has an inverse function if there is no horizontal line that intersects the graph of the function f at more than one point. 13a. Yes, the function has an 5. 5 inverse. 13b. No, the function does not 6. 8x have an inverse because it does not pass the horizontal line test. 13c. Yes, the function has an 7. x = 64 inverse it passes the horizontal line test x = x = 1/

8 See chapter 1, section x = log3 log log x 5.x = log8.7 log10 = log ln5 53. x = log6.1 loge log 5 y 35. log b x log b (z + 1) ( ) 36. log 3 x + log 3 x 1 ( ) 5 ( ) 37. log b x 3 y 38. ln y x ( ) x = log0 log log3.7 log x = log4 56. x = log4 log x log 57. x = 7 1 b y x 3 y x = = log y 59. x = e x = 60. x = x = x = 43. x = 1 6. x = e x = 1 63a. $77, x = 1 63b. $77, c. $78, million people

9 1. f( x) = x, Horizontal Asymptote is the x-axis or y = 0 (y = 0 is the equation of the x-axis) Y-intercept is (1, 0) x y -1 1/ g( x) = 1 x, H.A. y = 0 (y = 0 is the equation of the x-axis) 3 Y-intercept is (1, 0) x y /3

10 3. f(x) = 1+ x. H.A. y = 1. Y-intercept is (0. ) x y -1 3/ f( x) = log x, There is no y-intercept. The graph does not cross the y-axis. x y 1/

11 5. g x ( ) = log 1 X y / 1 x. Note that the graph will not cross the y-axis. 6. g(x) = 3 x+. Shift left two units. Horizontal asymptote does not move it remains y = 0. Y-intercept is (0, 9). x g (x) -3 1/

12 11.