Math 1101 Exam 3 Practice Problems These problems are not intended to cover all possible test topics. These problems should serve as an activity in preparing for your test, but other study is required to fully prepare. These problems contain some multiple choice questions, please consult with your instructor for particular details about your class test. 1. Decide whether or not the functions are inverses of each other. f(x) = 5 4x + 5, g(x) = x + 4 x Solution: ( ) 4x + 5 f(g(x)) = f x 5 x ) = 4x+5 + 4( x x 5x = 4x + 5 + 4x = 5x 8x + 5 Since the composite is not equal to x, they are not inverses of each other! You may also check the graphs of each function to determine if they are reflections over the line y = x. 2. Determine whether or not the function is one-to-one. (a) A. Yes B. No
MATH 1101 Exam 3 Review, Page 2 of 12 Fall 2013 (b) f(x) = 8x 2 3 A. Yes B. No 3. Determine whether the function that pairs students ID numbers with their GPAs is an invertible function. A. Yes B. No 4. Find the inverse of the function f(x) = 5x 2 2x 8 Solution: y = 5x 2 2x 8 x = 5y 2 2y 8 x(2y 8) = 5y 2 2xy 8x = 5y 2 2xy 8x 5y = 2 2xy 5y = 8x 2 y(2x 5) = 8x 2 y = 8x 2 2x 5 f 1 (x) = 8x 2 2x 5 5. Graph the function using transformations of an exponential function s graph with which you are familiar f(x) = 4 x 2 + 3 Solution: The graph of f(x) is obtained by shifting the graph of g(x) = 4 x horizontally 2 units to the right and vertically 3 units upward.
MATH 1101 Exam 3 Review, Page 3 of 12 Fall 2013 6. A computer is purchased for $4800. Its value each year is about 78% of the value the preceding year. Its value, in dollars, after t years is given by the exponential function V (t) = 4800(0.78) t. Find the value of the computer after 7 years. Solution: V (7) = 4800(0.78) 7 = 843.15, so the value is $843.15 after 7 years. 7. What are the domain and range for the equation y = 2 x? A. Domain: (0, ); Range (, ) B. Domain: (, ); Range (0, ) C. Domain: (, ); Range (, ) D. Domain: (, ); Range [0, ) 8. Write the logarithmic equation in exponential form. (a) log 5 125 = 3 Solution: 5 3 = 125 (b) ln x = 7 A. x = e 7 B. x = ln 7 C. No solution D. x = e 7 9. Find the value of without using a calculator. A. 6 B. 2 C. 2 D. 6 log 6 1 36
MATH 1101 Exam 3 Review, Page 4 of 12 Fall 2013 10. Graph the function f(x) = log 2 (x) + 4. Solution: The graph of f(x) is obtained by shifting the graph of g(x) = log 2 x vertically 4 units upward. 11. Find the inverse of the function f(x) = 8 x. A. f 1 (x) = log x 8 B. f 1 (x) = ln 8x C. f 1 (x) = log 8 x D. f 1 (x) = x 8 12. Solve the equation log 5 x = 2 1 A. 3 B. C. 10 D. 32 1 25 13. A certain noise( has an intensity I of 8.17 10 5. Give that decibel level L is related to intensity I by L = 10 log I 0 ), where I 0 = 10 12, determine the decibel level of the noise. Round your answer to the nearest decibel. A. 69 decibels B. 182 decibels C. 79 decibels D. 8 decibels 14. Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible. log 16 3 11 y 2 x
MATH 1101 Exam 3 Review, Page 5 of 12 Fall 2013 Solution: log 16 3 11 y 2 x = log 16 3 11 log16 (y 2 x) = 1 3 log 16 11 (log 16 y 2 + log 16 x) = 1 3 log 16 11 (2 log 16 y + log 16 x) = 1 3 log 16 11 2 log 16 y log 16 x 15. Rewrite as a single logarithm. 1 2 log 2 x 4 + 1 4 log 2 x 4 1 6 log 2 x A. log 2 x 17/6 B. log 2 x 9/2 C. 7 6 log 2 x 8 D. log 2 x 7 16. Solve the equation by hand. Give the exact answer and then an approximation rounded to the nearest thousandth. 5 5x 1 = 23 Solution: 5 5x 1 = 23 log 5 5 5x 1 = log 5 23 (5x 1) log 5 5 = log 5 23 5x 1 = log 5 23 5x = log 5 23 + 1 x = log 5 23 + 1 5 x 0.5896 17. Solve the equation. Give an exact solution. log(x 3) = 1 log x A. 5 B. 5, 2 C. 5 D. 2, 5
MATH 1101 Exam 3 Review, Page 6 of 12 Fall 2013 18. The sales of a new model of notebook computer are approximated by S(x) = 4000 14000e x/9, where x represents the number of months the computer has been on the market, and S represents the sales in thousands of dollars. In how many months will the sales reach $15,000? Solution: We would like to find the solution to 4000 14000e x/9 = 1500, one can do that in Graph by plotting both the left and right hand sides of the equation simultaneously and finding the intersection of the resulting graphs: This gives an answer of 15.5 months. 19. Assume the cost of a car is $27,000. With continuous compounding in effect, the cost of the car will increase according to the equation C = 27000e rt, where r is the annual inflation rate and t is the number of years. Find the number of years it would take to double the cost of the car at an annual inflation rate of 5.2%. Round the answer to the nearest hundredth. A. 209.55 years B. 1.96 years C. 13.33 years D. 196.22 years
MATH 1101 Exam 3 Review, Page 7 of 12 Fall 2013 20. Find the exponential function f that models this data. Round the coefficients to the nearest hundredth. A. f(x) = (1.09)(527.34) x B. f(x) = (567.57)(0.17) x C. f(x) = (527.34)(1.09) x D. f(x) = (0.17)(567.57) x x 1 2 3 4 y 580 620 670 750 21. Find the exponential function which models a solid with initial mass of 22 grams, decreasing at a rate of 3.1% per day. Solution: An exponential function which models growth or decay rate of r percent with an initial value of P 0 is given by P (t) = P 0 (1 + r) t. In this case, r = 0.031 and P 0 = 22 the function is: P (t) = 22(0.969) t 22. Find an exponential function that models the data below and use it to predict about how many books will have been read in the eighth grade. A. 330 B. 4496 C. 1883 D. 788 Grade Number of Books Read 2 9 3 27 4 67 5 121
MATH 1101 Exam 3 Review, Page 8 of 12 Fall 2013 23. Several years ago, a large city undertook a major effort to encourage carpooling in order to reduce traffic congestion. The accompanying table shows the number of carpoolers, in thousands, from 1995 to 2000. Use regression to obtain a function f(x) = a + b ln x that models the data, where x = 1 corresponds to 1995, x = 2 to 1996, and so on. Round the constants a and b to the nearest hundredth. Year 1995 1996 1997 1998 1999 2000 Carpoolers 3.9 8.1 10.3 11.8 12.6 13.2 A. f(x) = 4.40 + 5.32 ln x B. f(x) = 4.33 + 5.17 ln x C. f(x) = 4.22 + 5.25 ln x D. f(x) = 4.29 + 5.14 ln x 24. Select an appropriate type of modeling function for the data shown in the graph. y x A. Linear B. Cubic C. Exponential
MATH 1101 Exam 3 Review, Page 9 of 12 Fall 2013 25. Does it appear that a linear model or an exponential model is the better fit for the data given in the table below? x y 2 5 4 7.5 6 9.8 8 12.3 Solution: x y 1st diff. % change 2 5 2.5 50% 4 7.5 2.3 30.7% 6 9.8 2.5 25.5% 8 12.3 Since the first differences are relatively constant, and the percent change is not, a linear model is the better fit for this data. 26. Jorge invested $2500 at 4% interest compounded semi-annually. In how many years will Jorge s investment have tripled? Round your answer to the nearest tenth of a year. Solution: If P 0 dollars are invested at a rate of r compounded k times a year, the value of the investment in t years is given by V (t) = P 0 (1 + r k )kt. In this case, P 0 = 2500, r = 0.04, and k = 2 giving the function ( V (t) = 2500 1 + 0.04 ) 2t = 2500(1.02) 2t 2 Jorge s investment triples once its value becomes $7500: In this case, t = 27.7 years. 7500 = 2500(1.02) 2t 3 = (1.02) 2t log 3 = log(1.02) 2t log 3 = 2t log 1.02 log 3 log 1.02 = 2t log 3 2 log 1.02 = t
MATH 1101 Exam 3 Review, Page 10 of 12 Fall 2013 27. State the degree of the polynomial and whether the leading coefficient is positive or negative. A. 4: Leading coefficient is negative. B. 5: Leading coefficient is negative. C. 4: Leading coefficient is positive. D. 5: Leading coefficient is positive. 28. State the degree and leading coefficient of the polynomial f(x) = 2(x + 9) 2 (x 9) 2 A. Degree 4; leading coefficient 1 B. Degree 4; leading coefficient 2 C. Degree 2; leading coefficient 2 D. Degree 2; leading coefficient 1 29. Predict the end behavior of the graph of the function f(x) = 1.48x 4 x 3 + x 2 8x + 4 A. Down on both sides B. Up on both sides C. Up on left side, down on right side D. Down on left side, up on right side
MATH 1101 Exam 3 Review, Page 11 of 12 Fall 2013 30. P (x) = x 3 + 27 2 x2 60x + 100, x 5 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of tires that must be sold to maximize profit. A. 5.5 hundred thousand B. 4 hundred thousand C. 5 hundred thousand D. 4.5 hundred thousand 31. Approximate the coordinates of each turning point accurate to two decimal places y = x 4 4x 3 + 4x + 6 Solution: To approximate the coordinates of the turning points, you should plot the function in Graph. Then you may use the trace to Snap to the extreme y-values. These yield local minima of ( 0.5321, 4.5544) and (2.8794, 9.2344) and a local maximum of (0.6527, 7.6800).
MATH 1101 Exam 3 Review, Page 12 of 12 Fall 2013 32. The table below gives the number of births, in thousands, to females over the age of 35 for a particular state every two years from 1970 to 1986 Year Births (thousands) 1970 42.5 1972 29.9 1974 36.0 1976 56.9 1978 71.1 1980 69.9 1982 57.2 1984 37.1 1986 25.9 Use technology to find the quartic function that is the best fit for this data, where x is the number of years after 1970. According to the model, how many births were to females over the age of 35 in this state in 1990? A. 108,868 B. 106,368 C. 101,318 D. 94,368