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college algebra graphs and models
College Algebra 7th Edition Robert F Blitzer - Solutions
Solve and graph the solution set on a number line: 2x - 3 4 IV 3x 4 +
If you are given the equation of a rational function, explain how to find the vertical asymptotes, if any, of the function’s graph.
Write an equation in point-slope form and slope-intercept form of the line passing through (−10, 3) and (−2, −5).
If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function’s graph.
a. If y = kx2, find the value of k using x = 2 and y = 64.b. Substitute the value for k into y = kx2 and write the resulting equation.c. Use the equation from part (b) to find y when x = 5.
Divide 737 by 21 without using a calculator. Write the answer as quotient + remainder divisor
a. If y = k/x, find the value of k using x = 8 and y = 12.b. Substitute the value for k into y = k/x and write the resulting equation.c. Use the equation from part (b) to find y when x = 3.
Rewrite 4 - 5x - x2 + 6x3 in descending powers of x.
If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?
If S = kA/P, find the value of k using A = 60,000, P = 40, and S = 12,000.
Use 2x³ 3x² 11x + 6 x - 3 to factor 2x³ 3x² 11x + 6 completely. 2x² + 3x - 2
The rational functionmodels the number of arrests, f(x), per 100,000 drivers, for driving under the influence of alcohol, as a function of a driver’s age, x.a. Graph the function in a [0, 70, 5] by [0, 400, 20] viewing rectangle.b. Describe the trend shown by the graph.c. Use the Zoom and trace
Use a graphing utility to graph y = 1/x2, y = 1/x4, and y = 1/x6 in the same viewing rectangle. For even values of n, how does changing n affect the graph of y = 1/xn?
Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.
In Exercises 122–125, determine whether each statement makes sense or does not make sense, and explain your reasoning.My graph of has vertical asymptotes at x = 1 and x = 2. y x - 1 (x - 1)(x - 2)
In Exercises 122–125, determine whether each statement makes sense or does not make sense, and explain your reasoning.I’ve graphed a rational function that has two vertical asymptotes and two horizontal asymptotes.
In Exercises 122–125, determine whether each statement makes sense or does not make sense, and explain your reasoning.The functionmodels the fraction of nonviolent prisoners in New York State prisons x years after 1980. I can conclude from this equation that over time the percentage of nonviolent
In Exercises 122–125, determine whether each statement makes sense or does not make sense, and explain your reasoning.As production level increases, the average cost for a company to produce each unit of its product also increases.
In Exercises 126–129, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of a rational function cannot have both a vertical asymptote and a horizontal asymptote.
In Exercises 126–129, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.It is possible to have a rational function whose graph has no y-intercept.
In Exercises 126–129, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of a rational function can have three vertical asymptotes.
In Exercises 126–129, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of a rational function can never cross a vertical asymptote.
Identify the graphs (a)–(d) in which y is a function of x. a. b.
Which of the following graphs (a)–(d) represent functions that have an inverse function? a. X b. X
In Exercises 130–133, write the equation of a rational function f(x) = p(x)/q(x) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the
Basic Car Rental charges $20 a day plus $0.10 per mile, whereas Acme Car Rental charges $30 a day plus $0.05 per mile. How many miles must be driven to make the daily cost of a Basic Rental a better deal than an Acme Rental?
In Exercises 130–133, write the equation of a rational function f(x) = p(x)/q(x) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the
In Exercises 130–133, write the equation of a rational function f(x) = p(x)/q(x) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the
In Exercises 130–133, write the equation of a rational function f(x) = p(x)/q(x) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the
In Exercises 100–103, determine whether each statement makes sense or does not make sense, and explain your reasoning.Although I have not yet learned techniques for finding the x-intercepts of f(x) = x3 + 2x2 - 5x - 6, I can easily determine the y-intercept.
Describe how to graph a rational function.
Disprove each statement in Exercises 116–120 bya. letting y equal a positive constant of your choice, andb. using a graphing utility to graph the function on each side of the equal sign. The two functions should have different graphs, showing that the equation is not true in general.ln(x - y) =
Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, f(t), after t months was modeled by the
The pH scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to 14. A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the pH below 7, the more acidic is the
Disprove each statement in Exercises 116–120 bya. letting y equal a positive constant of your choice, andb. using a graphing utility to graph the function on each side of the equal sign. The two functions should have different graphs, showing that the equation is not true in general.ln(xy) = (ln
Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.
The pH scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to 14. A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the pH below 7, the more acidic is the
What question can be asked to help evaluate log3 81?
Explain how to solve an exponential equation when both sides can be written as a power of the same base.
In Exercises 121–124, determine whether each statement makes sense or does not make sense, and explain your reasoning.Because I cannot simplify the expression bm + bn by adding exponents, there is no property for the logarithm of a sum.
Explain why the logarithm of 1 with base b is 0.
Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use 3x = 140 in your explanation.
In Exercises 121–124, determine whether each statement makes sense or does not make sense, and explain your reasoning.Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.
Describe the following property using words: logb bx = x.
Explain the differences between solving log3(x - 1) = 4 and log3(x - 1) = log3 4.
In Exercises 121–124, determine whether each statement makes sense or does not make sense, and explain your reasoning.I can use any positive number other than 1 in the change of- base property, but the only practical bases are 10 and e because my calculator gives logarithms for these two bases.
Explain how to use the graph of f(x) = 2x to obtain the graph of g(x) = log2 x.
In Exercises 121–124, determine whether each statement makes sense or does not make sense, and explain your reasoning.I expanded by writing the radical using a rational exponent and then applying the quotient rule, obtaining 1/2 log4 x - log4 y. log4 X y
In many states, a 17% risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the 17% risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your
The function P(x) = 95 - 30 log2 x models the percentage, P(x), of students who could recall the important features of a classroom lecture as a function of time, where x represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the
Write as a single term that does not contain a logarithm: eln 8x³ - In 2x²
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.3x = 2x + 3
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. log7 49 log77 log749 - log77
In Exercises 128–131, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f.f(x) = log x, g(x) = log(x - 2) + 1
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.log(x - 15) + log x = 2
In Exercises 128–131, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f.f(x) = log x, g(x) = -log x
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In V2 = In 2 2
If log 3 = A and log 7 = B, find log7 9 in terms of A and B.
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.log3(3x - 2) = 2
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.logb(x3 + y3) = 3 logb x + 3 logb y
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.log3(4x - 7) = 2
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.2x+1 = 8
Suppose that a girl is 4 feet 6 inches at age 10. Explain how to use the function in Exercises 113–114 to determine how tall she can expect to be as an adult.Data from exercise 113-114The percentage of adult height attained by a girl who is x years old can be modeled by f(x) = 62 + 35 log(x - 4),
Explain how to find the domain of a logarithmic function.
If f(x) = logb x, show that f(x +h)-f(x) h = h 1080 (1 141) ².1 + log + X h = 0.
Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, f(t), for the group after t months was modeled by the human memory function f(t) = 75 - 10 log(t + 1), where 0 ≤ t ≤ 12. Use a graphing utility to
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.5x = 3x + 4
In parts (a)–(c), graph f and g in the same viewing rectangle.a. f(x) = ln(3x), g(x) = ln 3 + ln xb. f(x) = log(5x2), g(x) = log 5 + log x2c. f(x) = ln(2x3), g(x) = ln 2 + ln x3d. Describe what you observe in parts (a)–(c). Generalize this observation by writing an equivalent expression for
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning.Because the equations log(3x + 1) = 5 and log(3x + 1) = log 5 are similar, I solved them using the same method.
In Exercises 135–138, determine whether each statement makes sense or does not make sense, and explain your reasoning.I can evaluate some common logarithms without having to use a calculator.
Exercises 137–139 will help you prepare for the material covered in the next section.Solve: x + 2 4x + 3 || 1 X
Givenfind each of the following:a. (f ° g)(x)b. The domain of f ° g. f(x) 2 x + 1 and g(x) 1 X
Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. 2 y = x, y = √x, y = e, y = ln x, y = x, y = x²
Exercises 137–139 will help you prepare for the material covered in the next section.138. Solve: x(x - 7) = 3.
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning.I can solve 4x = 15 by writing the equation in logarithmic form.
Exercises 137–139 will help you prepare for the material covered in the next section.Solve for x: a(x - 2) = b(2x + 3).
In Exercises 135–138, determine whether each statement makes sense or does not make sense, and explain your reasoning.An earthquake of magnitude 8 on the Richter scale is twice as intense as an earthquake of magnitude 4.
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning.Because the equations 2x = 15 and 2x = 16 are similar,bI solved them using the same method.
Use the Leading Coefficient Test to determine the end behavior of the graph of f(x) = -2x2(x - 3)2(x + 5).
Hurricanes are one of nature’s most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function f(x) = 0.48 ln(x + 1) + 27 models the barometric air pressure, f(x), in inches of mercury, at a distance of x miles from the eye of a hurricane. Use this function
In Exercises 135–138, determine whether each statement makes sense or does not make sense, and explain your reasoning.I’ve noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential
Hurricanes are one of nature’s most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function f(x) = 0.48 ln(x + 1) + 27 models the barometric air pressure, f(x), in inches of mercury, at a distance of x miles from the eye of a hurricane. Use this function
In Exercises 139–142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.log(-100) = -2
In Exercises 125–132, use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the x-coordinate of the intersection point to find the equation’s solution set. Verify this value by direct substitution into the equation.log(x + 3) + log x = 1
In Exercises 128–131, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f.f(x) = ln x, g(x) = ln x + 3
Use the change-of-base property to prove that log e = 1 In 10
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.logb(xy)5 = (logb x + logb y)5
In Exercises 128–131, graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f.f(x) = ln x, g(x) = ln(x + 3)
The function P(t) = 145e-0.092t models a runner’s pulse, P(t), in beats per minute, t minutes after a race, where 0 ≤ t ≤ 15. Graph the function using a graphing utility. trace along the graph and determine after how many minutes the runner’s pulse will be 70 beats per minute. Round to the
The function W(t) = 2600(1 - 0.51e-0.075t)3 models the weight, W(t), in kilograms, of a female African elephant at age t years. (1 kilogram ≈ 2.2 pounds) Use a graphing utility to graph the function. Then trace along the curve to estimate the age of an adult female elephant weighing 1800
Graph: f(x) = 4x2/ x2 - 9
In Exercises 139–142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. log₂8 log2 4 814
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning.It’s important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.
In Exercises 141–144, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.If log(x + 3) = 2, then e2 = x + 3.
In Exercises 9–20, write each equation in its equivalent logarithmic form. 5-3 || 1 125
In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. In e2 5
In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. (125) logs
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents.125x = 625
In Exercises 10–20, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log100 10
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 31-x= 27
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