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linear algebra

Questions and Answers of Linear Algebra

Foreign-Born Population. In 1970, only 4.7% of the U.S. population was foreign-born, while in 2007, 12.6% of the population was foreign-born (Sources: Annual Social and Economic Supplements, Current
Sketch the graph of the polynomial functiona. f (x) = x3 - 5x2 + 2x + 8b. f (x) = -2x4 + x3 + 11x2 - 4x - 12
Using the intermediate value theorem, determine, if possible, whether the function has a zero between a and b. a. f (x) = -5x2 + 3; a = 0, b = 2 b. g(x) = 2x3 + 6x2 - 3; a = -2, b = -1
Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x). Express P(x) in the form d(x) ( Q(x) + R(x). Show your work. P(x) = x4 + 3x3 + 2x - 5, d(x) = x - 1
Find the inverse of the relation. a. {(7, 8), (-2, 8), (3, -4), (8, -8)} b. {(0, 1), (5, 6), (-2, -4)} c. {(-1, -1), (-3, 4)} d. {(-1, 3), (2, 5), (-3, 5), (2, 0)}
The total amount of spending per year, in billions of dollars, on pets in the United States x years after 2000 is given by the function P(x) = 2.1782x + 25.3 (Source: Animal Pet Products
The following formula can be used to convert Fahrenheit temperatures x to Celsius temperatures T(x): T(x) = 5/9 (x - 32). a) Find T (-13o) and T (86o). b) Find T-1(x) and explain what it represents.
A function that will convert women's shoe sizes in the United States to those in Australia is S(x) = (2x - 3) / 2 (Source: OnlineConversion.com). a) Determine the women's shoe sizes in Australia that
Consider the following quadratic functions. Without graphing them, answer the questions below.a) f(x) = 2x2b) f(x) = -x2c) f(x) = ¼ x2d) f(x) = -5x2 + 3e) f(x) = 2/3(x - 1)2 - 3f) f(x) = -2(x + 3)2
Using only a graphing calculator, determine whether the functions are inverses of each other.a. f(x) = 3√((x - 3.2) / 1.4) , g(x) = 1.4x3 + 3.2b. f (x) = (2x - 5) / (4x + 7), g(x) = (7x - 4) / (5x
Graph the equation by substituting and plotting points. Then reflect the graph across the line y = x to obtain the graph of its inverse. a. x = y2 - 3 b. y = x2 + 1 c. y = 3x - 2
Given the function f, prove that f is one-to-one using the definition of a one-to-one function. a. f (x) = 1/3 x - 6 b. f (x) = 4 - 2x c. f (x) = x3 + 1/2
Given the function g, prove that g is not one-to-one using the definition of a one-to-one function. a. g(x) = 1 - x2 b. g(x) = 3x2 + 1 c. g(x) = x4 - x2 d. (1x) = 1/x6
Using the horizontal-line test, determine whether the function is one-to-one.a. f(x) = 2.7xb. f (x) = 2-x c. f(x) = 4 - x2 d. f(x) = x3 - 3x + 1
Graph the function and determine whether the function is one-to-one using the horizontal-line test. a. f(x) = 5x - 8 b. f(x) = 3 + 4x c. f(x) = 1 - x2
Graph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of f and of f -1. a. f(x) = 0.8x + 1.7 b. f(x) = 2.7 -
Find an equation of the inverse relation. a. y = 4x - 5 b. 2x2 + 5y2 = 4 c. x3y = -5 d. y = 3x2 - 5x + 9
In Exercises, for each function: a) Determine whether it is one-to-one. b) If the function is one-to-one, find a formula for the inverse. 1. f(x) = x + 4 2. f(x) = 7 - x
Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.
Each graph in Exercises is the graph of a one-to-one function f. Sketch the graph of the inverse function f-1.a.b. c.
For the function f, use composition of functions to show that f-1 is as given. a. f(x) = 7/8 x, f-1(x) = 8/7 x b. f(x) = (x + 5) / 4, f -1(x) = 4x - 5 c. f(x) = (1 - x)/x, f -1(x) = 1/(x + 1)
Find the inverse of the given one-to-one function f. Give the domain and the range of f and of f-1, and then graph both f and f-1 on the same set of axes. a. f(x) = 5x - 3 b. f(x) = 2 - x c. f(x) =
A city swimming league determines that the cost per person of a group swim lesson is given by the formula C(x) = (60 + 2x) / x, Where x is the number of people in the group and C(x) is in dollars.
Find the x-intercepts and the zeros of the function. a. f (x) = 2x2 - 13x - 7 b. h (x) = x3 - 3x2 + 3x - 1
Which is larger, 7π or π7? 7080 or 8070?
Graph the function by substituting and plotting points. Then check your work using a graphing calculator.a. f(x) = 3xb. f(x) = 5x
a) Graph using a graphing calculator. b) Approximate the zeros. c) Approximate the relative maximum and minimum values. If your graphing calculator has a MAX-MIN feature, use it. 1. f(x) = x2e-x 2.
Graph f (x) = x1/(x-1) for x > 0. Use a graphing calculator and the TABLE feature to identify the horizontal asymptote.
For the function f, construct and simplify the difference quotient. f (x) = 2ex - 3
Sketch the graph of the function and check the graph with a graphing calculator. Describe how each graph can be obtained from the graph of a basic exponential function. a. f(x) = 2x + 1 b. f(x) = 2x
In Exercises, match the function with one of the graphs (a)-(f), which follow.a.b. c. d. e. f. 1. f(x) = -2x - 1 2. f(x) = - (1/2)x 3. f(x) = ex + 3 4. f(x) = ex+1
Suppose that $82,000 is invested at 4 1/2, interest, compounded quarterly. a) Find the function for the amount to which the investment grows after t years. b) Graph the function. c) Find the amount
Suppose that $750 is invested at 7% interest, compounded semiannually. a) Find the function for the amount to which the investment grows after t years. b) Graph the function. c) Find the amount of
On Will's sixth birthday, his grandparents present him with a $3000 certificate of deposit (CD) that earns 5% interest, compounded quarterly. If the CD matures on his sixteenth birthday, what amount
Following the birth of her child, Sophia deposits $10,000 in a college trust fund where interest is 3.9%, compounded semiannually. a) Find a function for the amount in the account after t years. b)
In Exercises, use the compound-interest formula to find the account balance A with the given conditions:P = principal,r = interest rate,n = number of compounding periods per year,t = time, in years,A
The number of U.S. troops diagnosed as overweight has more than doubled since 1998 (Source: U.S. Department of Defense). The exponential function W(x) = 23,672.161 (.1122)x Where x is the number of
The number of students from India enrolled in U.S. colleges and universities has grown exponentially from about 9000 in 1980 to about 95,000 in 2008 (Source: Institute of International Education, New
The number of pages in the IRS Tax Code has increased from 400 pages in 1913 to 70,320 pages in 2009 (Source: IRS). The increase can be modeled by the exponential function T(x) = 400 (1.055)x Where x
The monthly Medicare Part B health-care premium for most beneficiaries ages 65 and older has increased significantly since 1975. The monthly premium has increased from about $7 in 1975 to $110.50 in
The business of taxi medallions, required licenses fastened to the hood of all New York City cabs has proven itself to be a valuable investment. The average price for a corporate-license taxi
The cost per household for taking the U.S. census has increased exponentially since 1970 (Source: GAO analysis of U.S. Census Bureau data). The following function models the increase in cost: C(x) =
It is estimated that 90 billion plastic carry-out bags are produced annually in the United States. In recent years, the number of tons of plastic bags that are recycled has grown exponentially from
The function D(t) = 43.1224(1.0475)t Gives the number of master's degrees, in thousands, conferred on women in the United States t years after 1960 (Sources: National Center for Educational
In recent years, U.S. tennis participation has increased. The function T(x) = 23.7624(1.0752)x, Where T(x) is in millions and x is the number of years since 2006, models the number of people who had
Over the last four decades, charitable giving in the United States has grown exponentially from approximately $20.7 billion in 1969 to approximately $303.8 billion in 2009 (Sources: Giving USA
A landscape company purchased a backhoe for $56,395. The value of the backhoe each year is 90% of the value of the preceding year. After t years, its value, in dollars, is given by the exponential
A restaurant purchased a 72-in. range with six burners for $6982. The value of the range each year is 85% of the value of the preceding year. After t years, its value, in dollars, is given by the
A company begins an Internet advertising campaign to market a new phone. The percentage of the target market that buys a product is generally a function of the length of the advertising campaign. The
The value of a stock is given by the function V(t) = 58 (1 - e-1.1t) + 20, Where V is the value of the stock after time t, in months a) Graph the function. b) Find V(1), V(2), V(4), V(6), and
In Exercises, use a graphing calculator to match the equation with one of the figures (a)-(n), which follow.1. y = 3x - 3-x 2. y = 3-(x+1)2 3. f (x) = -2.3x 4. f (x) = 30,000(1.4)x 5. y = 2-|x|
Use a graphing calculator to find the point(s) of intersection of the graphs of each of the following pairs of equations. a. y = |1 - 3x|, y = 4 + 3-x2 b. y = 4x + 4-x, y = 8 - 2x - x2 c. y = 2ex -
Make a hand-drawn graph of each of the following. Then check your work using a graphing calculator.a. x = 3yb. x = 4y
A model for advertising response is given by the function N(a) = 1000 + 200 ln a, a ≥ 1, Where N(a) is the number of units sold when a is the amount spent on advertising, in thousands of dollars.
The loudness L, in bels (after Alexander Graham Bell), of a sound of intensity I is defined to be L = log I/I0, Where I0 is the minimum intensity detectable by the human ear (such as the tick of a
Find the slope and the y-intercept of the line. a. 3x - 10y = 14 b. y = 6 c. x = -4
Use synthetic division to find the function values. a. g(x) = x3 - 6x2 + 3x + 10; find g(-5) b. f (x) = x4 - 2x3 + x - 6; find f (-1)
Find a polynomial function of degree 3 with the given numbers as zeros. Answers may vary. a. √7, -√7, 0 b. 4i, -4i, 1
Find the domain of the function. a. f (x) = log5 x3 b. f (x) = log4 x2 c. f (x) = ln | x |
In Exercises, match the equation with one of the figures (a)-(d), which follow.1. f (x) = ln | x | 2. f (x) = ln x2 3. f (x) = | ln x | 4. g(x) = | ln (x - 1)
a) Graph the function. b) Estimate the zeros. c) Estimate the relative maximum values and the relative minimum values. 1. f (x) = x ln x 2. f (x) = x2 ln x
Using a graphing calculator, find the point(s) of intersection of the graphs of the following. y = 4 ln x, y = 4 / (ex + 1)
Convert to a logarithmic equation. a. 103 = 1000 b. 5-3 = 1/125 c. 81/3 = 2
Convert to an exponential equation. a. log5 5 = 1 b. t = log4 7 c. log 0.01 = -2
Find the logarithm using common logarithms and the change-of-base formula. a. log4 100 b. log3 20 c. log100 0.3 d. log π 100
Find the logarithm using natural logarithms and the change-of-base formula. a. log3 12 b. log4 25 c. log100 15 d. log9 100
Graph the function and its inverse using the same set of axes. Use any method. a. f (x) = 3x, f -1(x) = log3 x b. f (x) = log4 x, f -1(x) = 4x c. f (x) = log x, f -1(x) = 10x
For each of the following functions, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then graph the function using a graphing calculator. Give the
Find each of the following. Do not use a calculator. a. log2 16 b. log3 9 c. log5 125 d. log2 64 e. log 0.001
Graph the piecewise function.a.b.
Various cities and their populations are given below. Find the average walking speed in each city. a) Seattle, Washington: 608,660 b) Los Angeles, California: 3,792,621 c) Virginia Beach, Virginia:
Students in a computer science class took a final exam and then took equivalent forms of the exam at monthly intervals thereafter. The average score S1t2, as a percent, after t months was found to be
Various locations of earthquakes and their intensities are given below. What was the magnitude on the Richter scale?a) San Francisco, California, 1906: 107.7 ˆ™ I0b) Chile, 1960: 109.5
In chemistry, the pH of a substance is defined aspH = -log[H+] ,Where H+ is the hydrogen ion concentration, in moles per liter find the pH of each substance.Substance Hydrogen ion Concentration a)
Find the hydrogen ion concentration of each substance, given the pH. Express the answer in scientific notation. SUBSTANCE pH a) Tap water......................7 b)
Find each of the following without a calculator. 1. log4 1 2. ln e-4/5 3. log 0.01 4. ln e2 5. ln 1
Find the logarithm using the change-of-base formula. a. log3 20 b. logπ 10
Explain why an even function f does not have an inverse f -1 that is a function.
Describe the differences between the graphs of f (x) = x3 and g(x) = 3x.
For each function, determine whether it is one-to-one, and if the function is one-to-one, find a formula for its inverse.a. f (x) = - 2 / xb. f (x) = 3 + x2
Given the function f (x) = √(x - 5), use composition of functions to show that f -1(x) = x2 + 5.
Given the one-to-one function f (x) = x3 + 2, find the inverse and give the domain and the range of f and of f -1.
Match the function with one of the graphs (a)-(h), which follow.1. y = log2 x 2. f (x) = 2x + 2 3. f (x) = ex-1 4. f (x) = ln x - 2 5. f (x) = ln (x - 2)
Express as a sum of logarithms. 1. log3 181 ∙ 272 2. log2 18 ∙ 642 3. log5 15 ∙ 1252 4. log4 164 ∙ 42
Suppose that loga x = 2. Find each of the following. a. loga (1/x) b. log1/a x
Write each of the following without using logarithms. a. loga x + loga y - mz = 0 b. ln a - ln b + xy = 0
Prove each of the following for any base a and any positive number x. loga (1/x) = -loga x = log1/a x loga (z + √(x2 - 5)/5) = -loga (x - √x2 - 5)
Express as a difference of logarithms. a. logt M / 8 b. loga 76 / 13 c. log x / y d. ln a / b e. ln r / s
Express in terms of sums and differences of logarithms. a. loga 6xy5z4 b. loga x3y2z c. logb (p2q5) / (m4b9) d. logb (x2y) / b3
Express as a single logarithm and, if possible, simplify. a. loga 75 + loga 2 b. log 0.01 + log 1000 c. log 10,000 - log 100 d. ln 54 - ln 6 e. ½ log n + 3 log m
Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round the answer to the nearest thousandth a. loga 2/11 b. loga 14 c. loga 98 d. loga 1/7
Given that logb 2 ≈ 0.693, logb 3 ≈ 1.099, and logb 5 ≈ 1.609, find each of the following, if possible round the answer to the nearest thousandth. a. logb 125 b. logb 53 c. logb 16 d. logb 30
In each of Exercises, classify the function as linear, quadratic, cubic, quartic, rational, exponential, or logarithmic. a. f (x) = 5 - x2 + x4 b. f (x) = 2x c. f (x) = - 3/4 d. f (x) = 4x - 8
Express as a single logarithm and, if possible, simplify. a. loga (x2 + xy + y2) + loga (x - y) b. loga (a10 - b10) - loga (a + b)
Express as a sum or a difference of logarithms.a. loga (x - y) / √(x2 - y2)b. loga √(9 - x2)
Given that loga x = 2, loga y = 3, and loga z = 4, find loga (4√(y2z5) / 4√(x3z-2).
Solve the exponential equation algebraically. Then check using a graphing calculator. a. 3x = 81 b. 2x = 32 c. 22x = 8 d. 37x = 27
Solve the logarithmic equation algebraically. Then check using a graphing calculator. a. log5 x = 4 b. log2 x = -3 c. log x = -4 d. log x = 1
Use a graphing calculator to find the approximate solutions of the equation. a. 2x - 5 = 3x + 1 b. 0.082e0.05x = 0.034 c. xe3x - 1 = 3
Approximate the point(s) of intersection of the pair of equations.a. 2.3x + 3.8y = 12.4, y = 1.1 ln 1x - 2.052b. y = ln 3x, y = 3x - 8 c. y = 2.3 ln 1x + 10.72, y = 10e-0.007(x)2

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