Math 140 Quiz 4 - Summer 2006 Solution Review (Small white numbers next to problem number represent its difficulty as per cent getting it wrong.)

Problem 1 (39) Graph f(x) as a solid line/curve & its inverse as a dashed line/curve on the same axes. f(x) = 4x. Recognize f(x) as a straight line of slope m = 4 and y-intercept (0,0). L Draw the line L: y = x. Construct the inverse function line by reflection in L. Or, solve x = 4y for y to get f- -1(x) = x/4. Then graph it. This is answer C).

Problem 2 (78) Decide whether or not f(x) and g(x) are inverses of each other with: a) f(x) = 3x + 5, g(x) = (1/3)x - 5; b) f(x) = (x - 6)2, g(x) = x1/2 + 6. One needs to see if: (go f )(x) = g(f(x)) = x or (fo g )(x) = f(g(x)) = x. a) f(g(x) ) = 3[(1/3)x - 5] + 5 = x - 10 => NO. b) f(g(x)) = [(x1/2 + 6) - 6]2 = [x1/2]2 = x => YES.

Problem 3 (09) Approximate each value using a calculator. Express answer rounded to three decimal places. e2.64 If necessary, use keystrokes 1, INV(or 2nd), ln to get value of e. Then, press yx key and enter 2.64 followed by equal (or enter) key. Result is 14.01320361..., which rounds to 14.013.

Problem 4 (30) Approximate each value using a calculator. Express answer rounded to three decimal places. log9(1/81) Without a calculator one notes, log9(1/81) = log9(9-2) = -2log9(9) = -2. Or, evaluate one, or other, change-of-base formula: log9(1/81) = log(1/81)/log(9), log9(1/81) = ln(1/81)/ln(9).

Problem 5 (09) Use a calculator to find the natural logarithm correct to four decimal places. ln(821/2) One notes, ln(821/2) = (1/2)ln(82) = 2.203359624 = 2.2034. Or, evaluate the square root first and find: ln(821/2) = ln(9.055385138) = 2.203359624 = 2.2034.

Convert to logarithmic form: 53 = 125. Problem 6 (30) Convert to logarithmic form: 53 = 125. One takes the base-5 logarithm of both sides to get log5(53) = log5(125). Then one observes that log5(53) = 3 log5(5) = 3. Thus, log5(125) = 3.

Problem 7 (30) If the following defines a one-to-one function, find the inverse of: f(x) = 2x + 8. The horizontal line test is satisfied since a graph of _ y = f(x) = 2x + 8 is a line. Hence, f(x) is 1-to-1. One proceeds by swapping x <=> y and solving for y. Thus, x = 2y + 8 => y = (x - 8)/2. This means that f -1(x) = (x - 8)/2. Check: f -1(f(x)) = [(2x + 8) - 8]/2 = x.

Problem 8 (35) If the following defines a one-to-one function, find the inverse of: f = {(-3, -7), (-2, -7), (-1, -1), (0, 2)}. Since both domain values –3 and –2 lead to –7, f fails the horizontal line test and is not 1-to-1. If one proceeded by swapping x <=> y in f , getting: f -1 = {(-7, -3), (-7, -2), (-1, -1), (2, 0)}? But this is not a function and would not check. For example, f -1(f(-3)) = f -1(-7) = -3 or -2 and not just -3.

Problem 9 (74) Compute the amount in m years if a principal P is invested at a nominal annual interest rate of r compounded as given. Round to the nearest cent. P = $480, m = 7, r = 7% compounded quarterly A = P (1 + r/n )nm = $480 (1 + 0.07/4 )(4)(7) = $480 (1.0175 )28 = $780.20

Problem 10 (74) How long will it take for an investment to triple in value if it earns 8.75% compounded continuously? Round your answer to three decimal places. A = Pert 3P = Pe0.0875t 3 = e0.075t ln(3) = ln(e0.0875t ) = 0.0875t ln(e) = 0.0875t 0.0875t = ln(3) => t = ln(3)/0.0875 = 12.556 yr

Problem 11 (43) Use a calculator and the base conversion formula to find the logarithm, correct to three decimal places. log6.9(5.5) Evaluate one, or other, change-of-base formula: log6.9(5.5) = log(5.5)/log(6.9) = 0.74036/0.83885 = 0.883, log6.9(5.5) = ln(5.5)/ln(6.9) = 1.70475/1.93152 = 0.883.

Select the matching exponential function to the graph. Problem 12 (35) Select the matching exponential function to the graph. Eliminate C), D), and E), choices since they are of form ax, a >1, which increases & is not shown. [y = 0.65-x = (1/.65)x = (1.538)x.] f(x) = ax = 0.65x Evaluate A) & B) for a test point, say x = -3, with yx on calculator. A): 0.32-3 = 30.5 is not on graph. B): 0.65-3 = 3.6 is on graph. Thus, the graph is of function f(x) = ax = 0.65x.

Select the matching logarithmic function to the graph. Problem 13 (43) Select the matching logarithmic function to the graph. Eliminate A), B), C), and E), choices by evaluating for a test point, say x = 1, with log on calculator & find a value that is not shown. Observe that the domain of f(x) in figure is x > 3. A) log(3 – 1) = 0.3 C) log(1) – 3 = – 3 E) log(1/3) = – .48 B) 3 – log(1) = 3 To verify D) is shown, recall known graph of log (x) with vertical asymptote at x = 0. It is now shifted to (h, k) = (3, 0) with a vertical asymptote at x = 3. f(x) = log (x - 3)

Solve the problem. log 5(25) = x Without a calculator one notes, x = log5(25) = log5(52) = 2 log5(5) = 2. Or, evaluate one, or other, change-of-base formula: log5(25) = log(25)/log(5), log5(25) = ln(25)/ln(5).

Solve the problem. log 5(x) = - 3 One computes the exponential function 5x of both sides (which must be equal because of the given equation) to get: 5log5(x) = 5-3 = 1/53 = 1/125. Then, one recalls that aloga(x) = x. Hence, x = 1/125.

Problem 16 (48) Solve the problem. pH = -log10[H+] Find the pH, if the [H+] = 5.2 x 10-3. Use your calculator. ___________ pH = -log10[5.2 x 10-3] ___________ = -(-2.284) _ __________ = 2.28

Problem 17 (43) The formula D = 6e-0.04h can be used to find the number of milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. When the number of milligrams reaches 2, the drug is to be given again. What is the time between injections? Rearrange 2 = 6e-0.04h ___________ 1/e-0.04h = 6/2 or e0.04h = 3 _____ ln(e0.04h ) = 0.04h ln(e) = 0.04h = ln(3) h = ln(3)/0.04 = 27.47 hours

Problem 18 (78) Find the value of the expression. Let logbA = 3 and logbB = -4. Find logb[(AB)1/2]. logb[(AB)1/2] = (1/2)logb(AB) = _ (1/2)[logb(A)(+ logb(B)] = (1/2)[ (3)(+ (-4) ] = - 1/2

Solve the equation. log 5x = log 4 + log (x - 3) Problem 19 (74) Solve the equation. log 5x = log 4 + log (x - 3) Compute the exponential function 10x of both sides (which must be equal because of the given equation) and use aloga(x) = x to get: 10log 5x = 10[log 4 +log(x –3)] = 10log 4 10log(x –3)] 5x = 4(x - 3) __________ 5x - 4x = - 12________ x = -12 Note: testing => log arguments, 5x= -60 & x-3= -15 are < 0; this is not allowed. Thus, there is no solution. ________

Solve the equation. logy 8 = 5 Problem 20 (39) Solve the equation. logy 8 = 5 Compute the exponential function yx of both sides (which must be equal because of the given equation) and use aloga(x) = x to get: ylogy8 = 8 = y5 y5 = 8 __________ y = 81/5________ {81/5 }