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mathematics
college algebra graphs and models
College Algebra 7th Edition Robert F Blitzer - Solutions
These exercises investigate the relationship between polynomial functions and their average rates of change. For example, the average rate of change of f(x) = x2 from x to x + 0.001 for any x can be calculated and graphed as shown in the figures. The graph of f is a parabola, and the graph of its
Determine whether f is a rational function and state its domain. f(x) = ± +1
Find any vertical or horizontal asymptotes for the graph ofState the domain of f. 4x² x² f(x): =-
Use the given zeros to write the complete factored form of f(x). f(x) = x³ = 2x² - 5x + 6; zeros: -2, 1, and 3
Determine if the function is a polynomial function. If it is, state its degree and leading coefficient a. f(x) = 5 - 4x
Use the given zeros to write the complete factored form of f(x). f(x) = 6x² + 21x = 90; zeros: -6 and -
Divide the expression. 2x43x² + 4x - 7
These exercises investigate the relationship between polynomial functions and their average rates of change. For example, the average rate of change of f(x) = x2 from x to x + 0.001 for any x can be calculated and graphed as shown in the figures. The graph of f is a parabola, and the graph of its
The graph and degree of a polynomial with real coefficients f(x) are given. Determine the number of real zeros and the number of nonreal complex zeros. Assume that all zeros of f(x) are distinct. Degree 4 15 5 -5 -10 I 3 y = f(x) X
Use the graph of the polynomial function f to complete the following. Let a be the leading coefficient of the polynomial f(x). (a) Determine the number of turning points and estimate any x-intercepts. (b) State whether a > 0 or a (c) Determine the minimum degree of f. -1 3 3
Solve the rational equation (a) Symbolically, (b) Graphically, and (c) Numerically. 1 I+x 1 x-1 1
Determine whether f is a rational function and state its domain. f(x) |x-1| x + 1
Sketch a graph of f(x) = -x(x + 1)2 by hand and identify the x- and y-intercepts.
In Exercises 95–96, let f and g be defined by the following table: x -2 -1 0 1 2 f(x) 6 3 -1 -4 0 g(x) 0 4 1 -3 -6
Show thatis its own inverse. f(x) 3x - 2 5r – 3
A telephone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.Plan A• $30 per month buys 120 minutes.• Additional time costs $0.30 per minute.Plan B• $40 per month buys 200 minutes.• Additional
Which graphs in Exercises 96–99 represent functions that have inverse functions? -4-3-2 -2-1 y -3- .............. 2.3 (DP²³▬ICIO TH X
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. g(x) = x³ - 2
Find the area of the donut-shaped region bounded by the graphs of (x - 2)2 + (y + 3)2 = 25 and (x - 2)2 + (y + 3)2 = 36.
In Exercises 95–96, find all values of x satisfying the given conditions.f(x) = 1 - 2x, g(x) = 3x2 + x - 1, and (f ° g)(x) = -5.
Explain how to graph the equation x = 2. Can this equation be expressed in slope-intercept form? Explain.
Exercises 101–103 will help you prepare for the material covered in the first section of the next chapter.0 = -2(x - 3)2 + 8
In Exercises 101–102, find an equation for f -1(x). Then graph f and f -1 in the same rectangular coordinate system.f(x) = 1 - x2, x ≥ 0
The bar graph shows the population of the United States, in millions, for seven selected years. Here are two functions that model the data:Use the functions to solve Exercises 97–98.a. Write a function that models the total U.S. population for the years shown in the bar graph.b. Use the function
The bar graph shows the population of the United States, in millions, for seven selected years. Here are two functions that model the data:Use the functions to solve Exercises 97–98.a. Write a function that models the difference between the female U.S. population and the male U.S. population for
Which graphs in Exercises 96–99 represent functions that have inverse functions? H 4-3-2 y ॐ प्रै Afx X
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. g(x) = (x 3)
In Exercises 97–98, find f(-x) - f(x) for the given function f. Then simplify the expression.f(x) = x3 + x - 5
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. g(x) = (x - 2)³
A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is x2 + y2 = 25 at the point (3, -4).
Which graphs in Exercises 96–99 represent functions that have inverse functions? -10 -3-2- y П X
Divide and express the result in standard form: 4i + 7 5 - 2i
In Exercises 97–98, write a piecewise function that models each telephone billing plan. Then graph the function.$50 per month buys 400 minutes. Additional time costs $0.30 per minute.
Freedom 7 was the spacecraft that carried the first American into space in 1961. Total flight time was 15 minutes and the spacecraft reached a maximum height of 116 miles. Consider a function, s, that expresses Freedom 7’s height, s(t), in miles, after t minutes. Is s a one-to-one function?
A company that sells radios has yearly fixed costs of $600,000. It costs the company $45 to produce each radio. Each radio will sell for $65. The company’s costs and revenue are modeled by the following functions, where x represents the number of radios produced and sold: C(x) = 600,000 +
Solve and determine whether the equation 7(x - 2) + 5 = 7x - 9 is an identity, a conditional equation, or an inconsistent equation.
Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
In Exercises 97–98, find f(-x) - f(x) for the given function f. Then simplify the expression.f(x) = x2 - 3x + 7
The Corruption Perceptions Index uses perceptions of the general public, business people, and risk analysts to rate countries by how likely they are to accept bribes. The ratings are on a scale from 0 to 10, where higher scores represent less corruption. The graph shows the corruption ratings for
If f(2) = 6, and f is one-to-one, find x satisfying 8 + f -1(x - 1) = 10.
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. EX- x = (x)y
In Exercises 97–98, write a piecewise function that models each telephone billing plan. Then graph the function.$60 per month buys 450 minutes. Additional time costs $0.35 per minute.
Which graphs in Exercises 96–99 represent functions that have inverse functions? -4-3- -2- HIIHII X
The Corruption Perceptions Index uses perceptions of the general public, business people, and risk analysts to rate countries by how likely they are to accept bribes. The ratings are on a scale from 0 to 10, where higher scores represent less corruption. The graph shows the corruption ratings for
Explain how to use intercepts to graph the general form of a line’s equation.
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.State the intervals on which the graph giving the percent body fat in women is increasing and
Use the graph of f in the figure shown to draw the graph of its inverse function. -4-3-2 32 y 3- -3- y = f(x) 2 3 4 X
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. h(x) = (x - 2)² 3
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.State the intervals on which the graph giving the percent body fat in men is increasing and
The bar graph shows your chances of surviving to various ages once you reach 60.The functionsmodel the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.a. Find and interpret f(70).b. Find and interpret g(70).c. Which function serves
Take another look at the scatter plot in Exercise 91. Although there is a relationship between literacy and child mortality, we cannot conclude that increased literacy causes child mortality to decrease. Offer two or more possible explanations for the data in the scatter plot.
Solve and graph the solution set on a number line: -9 ≤ 4x - 1 < 15.
In Tom Stoppard’s play Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas
A department store has two locations in a city. From 2012 through 2016, the profits for each of the store’s two branches are modeled by the functions f(x) = -0.44x + 13.62 and g(x) = 0.51x + 11.14. In each model, x represents the number of years after 2012, and f and g represent the profit, in
Use a graphing utility to graph each equation in Exercises 100–103.Then use the .Trace. feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.y = 2x + 4
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. h(x) = x³
Use a graphing utility to graph each equation in Exercises 100–103.Then use the .Trace. feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation. y = -x-5
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.For what age does the percent body fat in women reach a maximum? What is the percent body fat for
The bar graph shows your chances of surviving to various ages once you reach 60.The functionsmodel the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.a. Find and interpret f(90).b. Find and interpret g(90).c. Which function serves
In Exercises 101–102, find an equation for f -1(x). Then graph f and f -1 in the same rectangular coordinate system. f(x) = √x + 1
Use a graphing utility to graph each equation in Exercises 100–103.Then use the .Trace. feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.y = -3x + 6
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. h(x) = x³ 4
Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7,2) and (x2, Y2) (1, -1). Find √(x₂ − x₁)² + (y₂ − y₁)². Express the answer in simplified - radical form.
The wage gap is used to compare the status of women’s earnings relative to men’s. The wage gap is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for women by the median annual earnings for men. The bar graph shows the wage gap for selected years
Exercises 101–103 will help you prepare for the material covered in the first section of the next chapter.-x2 - 2x + 1 = 0
Use a graphing utility to graph each equation in Exercises 100–103.Then use the .Trace. feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation. 3 y = x - 2
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.At what age does the percent body fat in men reach a maximum? What is the percent body fat for that
The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x.a. Describe what the functions f and g model in terms of the price of the computer.b. Find (f ° g)(x) and describe what this models in terms of the price of the computer.c. Repeat part (b) for (g f)(x).d. Which
Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries.a. Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table.b. Use the scatter plot capability to draw a scatter
Solve and graph the solution set on a number line:3|2x - 1 ≥ 21.
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. r(x) = (x 3)³ + 2
Exercises 101–103 will help you prepare for the material covered in the first section of the next chapter.Use the graph of f(x) = x2 to graph g(x) = (x + 3)2 + 1.
The regular price of a pair of jeans is x dollars. Let f(x) = x - 5 and g(x) = 0.6x.a. Describe what functions f and g model in terms of the price of the jeans.b. Find (f ° g)(x) and describe what this models in terms of the price of the jeans.c. Repeat part (b) for (g ° f)(x).d. Which composite
The wage gap is used to compare the status of women’s earnings relative to men’s. The wage gap is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for women by the median annual earnings for men. The bar graph shows the wage gap for selected years
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.Use interval notation to give the domain and the range for the graph of the function for women.
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.The function p(x) = -0.002x2 + 0.15x + 22.86 models percent body fat, p(x), where x is the number of
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. r(x) = (x - 2) + 1
In Exercises 103–104, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.(-2, 3) and (3, -9)
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.Use interval notation to give the domain and the range for the graph of the function for men. Percent
If a function is defined by an equation, explain how to find its domain.
Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.
In Exercises 103–104, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.(-4, 3) and (-2, 5)
If equations for f and g are given, explain how to find f - g.
Solve by completing the square: y2 - 6y - 4 = 0.
If equations for two functions are given, explain how to obtain the quotient function and its domain.
A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret C(90).
In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.The graph of my linear function at first increased, reached a maximum point, and then decreased.
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. h(x)=1/(x-3)³ - 2
In Exercises 105–106, find the midpoint of each line segment with the given endpoints.(2, 6) and (-12, 4)
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.The function p(x) = -0.004x2 + 0.25x + 33.64 models percent body fat, p(x), where x is the number of
Describe a procedure for finding (f ° g)(x). What is the name of this function?
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for a recent year.The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and T(x) is the tax owed:Use this information to solve Exercises
In Exercises 95–106, begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. h(x) = (x - 2)³ - 1
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for a recent year.The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and T(x) is the tax owed:Use this information to solve Exercises
In Exercises 105–108, you will be developing functions that model given conditions.You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the
In Exercises 105–108, you will be developing functions that model given conditions.A car was purchased for $22,500. The value of the car decreased by $3200 per year for the first six years. Write a function that describes the value of the car, V, after x years, where 0 ≤ x ≤ 6. Then find and
In Exercises 105–106, find the midpoint of each line segment with the given endpoints.(4, -6) and (-15, 2)
In Exercises 107–118, begin by graphing the cube root function, f(x) = 3√x. Then use transformations of this graph to graph the given function. g(x)=√x + 2
In Exercises 107–118, begin by graphing the cube root function, f(x) = 3√x. Then use transformations of this graph to graph the given function. g(x) = √x - 2
In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.A linear function that models tuition and fees at public four year colleges from 2000 through 2016 has negative slope.
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