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mathematics
linear algebra

Questions and Answers of Linear Algebra

A container firm is designing an open-top rectangular box, with a square base, that will hold 108 in3. Let x = the length of a side of the base.(a) Express the surface area as a function of x. (b)
Graph each of the following.(a)(b)
For the function in Exercise 18, find f(-1), f(5), f(-2), and f(-3).
For the function in Exercise 19, find f(-2), f(-1), f(0), and f(4).
Given that f(x) = √x - 2 and g(x) = x2 - 1, find each of the following, if it exists. (a) (f - g) (6) (b) (fg) (2) (c) (f + g) (-1)
For each pair of functions in Exercises: (1) f(x) = 4 / x2 ; g(x) = 3 - 2x (2) f(x) = 3x2 + 4x; g(x) = 2x - 1 (a) Find the domain of f, g, f + g, f - g, fg, and f/g. (b) Find (f + g) (x), (f - g)
Given the total-revenue and total-cost functions R(x) = 120x - 0.5x2 and C(x) = 15x + 6, find the total-profit function P(x).
For each function f, construct and simplify the difference quotient. (a) f(x) = 2x + 7 (b) f(x) = 3 - x2 (c) f(x) = 4 / x
Given that f(x) = 2x - 1, g(x) = x2 + 4, and h(x) = 3 - x3, find each of the following (a) (fog) (1) (b) (gof) (1) (c) (hof) (-2)
In Exercises for the pair of functions: (a) Find (fog) (x) and (fof) (x). (b) Find the domain of fog and the domain of gof. (1) f(x) = 4 / x2; g(x) = 3 - 2x (2) f(x) = 3x2 + 4x; g(x) = 2x - 1
Graph the given equation and determine visually whether it is symmetric with respect to the x-axis, the y-axis, and the origin. Then verify your assertion algebraically. (a) x2 + y2 = 4 (b) y2 = x2 +
Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.(1)(2)
Determine whether the function is even, odd, or neither even nor odd. (a) f(x) = 9 - x2 (b) f(x) = x3 - 2x + 4 (c) f(x) = x7 - x5
The shape of y = √x, but reflected across the x-axis and shifted right 3 units and up 4 units
The shape of y = |x|, but stretched vertically by a factor of 2 and shifted right 3 units
A graph of y = f(x) is shown below. No formula for f is given. Graph each of the following.(a) y = f(x - 1) (b) y = f(2x) (c) y = -2f(x)
Find an equation of variation for the given situation. (a) y varies directly as x, and y = 100 when x = 25. (b) y varies directly as x, and y = 6 when x = 9. (c) y varies inversely as x, and y = 100
Graph the function. Estimate the intervals on which the function is increasing or decreasing, and estimate any relative maxima or minima. (a) f(x) = x2 - 1 (b) f(x) = 2 - | x |
The time t required to empty a tank varies inversely as the rate r of pumping. If a pump can empty a tank in 35 min at the rate of 800 kL/min, how long will it take the pump to empty the same tank at
The score N on a test varies directly as the number of correct responses a. Ellen answers 29 questions correctly and earns a score of 87. What would Ellen's score have been if she had answered 25
The power P expended by heat in an electric circuit of fixed resistance varies directly as the square of the current C in the circuit. A circuit expends 180 watts when a current of 6 amperes is
For f(x) = x + 1 and g(x) = √x, the domain of (gof) (x) is which of the following? (a) [- 1, ∞) (b) [-1, 0) (c) [0, ∞) (d) (- ∞, ∞)
The graph of the function f is shown below.The graph of g(x) = - 1/2 f(x) + 1 is which of the following? a. b. c. d.
Prove that the sum of two odd functions is odd.
Given that f(x) = 4x3 - 2x + 7, find each of the following. Then discuss how each expression differs from the other. (a) f(x) + 2 (b) f (x + 2) (c) f(x) + f(2)
Given the graph of y = f(x), explain and contrast the effect of the constant c on the graphs of y = f(cx) and y = cf(x).
Describe conditions under which you would know whether a polynomial functionis even or odd without using an algebraic procedure. Explain.
Use a graphing calculator to find the intervals on which the function is increasing or decreasing, and find any relative maxima or minima. (a) f(x) = x2 - 4x + 3 (b) f(x) = -x2 + x + 6 (c) f(x) = x3
Determine the intervals on which the function is(a) increasing(b) decreasing(c) constant.
For f(x) = x2 and g(x) = √ x - 3, find each of the following. (a) The domain of f (b) The domain of g (c) The domain of f + g
Graph the function f(x) = 2 - x2. Estimate the intervals on which the function is increasing or decreasing, and estimate any relative maxima or minima.
For each function, construct and simplify the difference quotient. (a) f(x) 1 / 2 x + 4 (b) f(x) = 2x2 - x + 3
Given that f(x) = x2 - 1, g(x) = 4x + 3, and h(x) = 3x2 + 2x + 4, find each of the following. (a) (go h) (2) (b) (fo g) (- 1) (c) (ho f) (1)
Find (fo g) (x) and (go f) (x).
Find the domain of (fo g) (x) and the domain of (go f) (x).
Use a graphing calculator to find the intervals on which the function f(x) = x3 + 4x2 is increasing or decreasing, and find any relative maxima or minima.
Determine whether the graph of y = x4 - 2x2 is symmetric with respect to the x-axis, the y-axis, and the origin.
Determine whether the function f(x) = 2x / x2 + 1 is even, odd, or neither even nor odd. Show your work.
Write an equation for a function that has the shape of y = x2, but shifted right 2 units and down 1 unit.
Write an equation for a function that has the shape of y = x2, but reflected across the x-axis and shifted left 2 units and up 3 units.
The graph of a function y = f(x) is shown below. No formula for f is given. Graph y = - 1 / 2 f(x).
Find an equation of variation where y varies jointly as x and the square of z and inversely as w, and y = 100 when x = 0.1, z = 10, and w = 5.
The stopping distance d of a car after the brakes have been applied varies directly as the square of the speed r. If a car traveling 60 mph can stop in 200 ft, how long will it take a car traveling
A softball team is designing a triangular pennant such that the height is 6 in. less than four times the length of the base b. Express the area of the pennant as a function of b.
If (-3, 1) is a point on the graph of y = f(x), what point do you know is on the graph of y = f(3x)?
Graph:
For the function in Exercise 5, find f(- 7 / 8), f(5), and f(-4).
Given that f(x) = x2 - 4x + 3 and g(x) = √3 - x, find each of the following, if it exists. (f + g) (-6) (f - g) (-1) (fg) (2)
Express the number in terms of i. a. √-3 b. √-21 c. √-25 d. √-100 e. -√-33
Simplify. Write answers in the form a + bi, where a and b are real numbers. a. (-5 + 3i) + (7 + 8i) b. (-6 - 5i) + (9 + 2i) c. (4 - 9i) + (1 - 3i) d. (7 - 2i) + (4 - 5i)
Write a slope-intercept equation for the line containing the point (3, -5) and perpendicular to the line 3x - 6y = 7.
Given that f (x) = x2 + 4 and g(x) = 3x + 5, find each of the following. a. The domain of f - g b. The domain of f / g c. (f - g) (x) d. (f/g) (2)
For the function f (x) = x2 - 3x + 4, construct and simplify the difference quotient (f (x + h) - f (x)) / h.
After declining between 1940 and 1980, the number of multigenerational American households has been increasing since 1980. The function h(x) = 0.012x2 - 0.583x + 35.727 can be used to estimate the
After declining between 1940 and 1980, the number of multigenerational American households has been increasing since 1980. The function h(x) = 0.012x2 - 0.583x + 35.727 can be used to estimate the
The number of TV channels that the average U.S. home receives has been soaring in recent years. The function t(x) = 0.16x2 + 0.46x + 21.36 can be used to estimate this number, where x is the number
The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds. Use this formula for Exercise. a. The Taipei 101 Tower, also known as the
The length of a rectangular poster is 1 ft more than the width, and a diagonal of the poster is 5 ft. find the length and the width.
One leg of a right triangle is 7 cm less than the length of the other leg. The length of the hypotenuse is 13 cm. Find the lengths of the legs.
One number is 5 greater than another. The product of the numbers is 36. Find the numbers.
One number is 6 less than another. The product of the numbers is 72. Find the numbers.
An open box is made from a 10-cm by 20-cm piece of tin by cutting a square from each corner and folding up the edges. The area of the resulting base is 96 cm2. What is the length of the sides of the
At the Glen Island Zoo, 170 m of fencing was used to enclose a petting area of 1750 m2. Find the dimensions of the petting area.
Find the dimensions of a Persian rug whose perimeter is 28 ft and whose area is 48 ft2.
The frame on a picture is 8 in. by 10 in. outside and is of uniform width. What is the width of the frame if 48 in2 of the picture shows?
State whether the function is linear or quadratic. a. f (x) = 4 - 5x b. f (x) = 4 - 5x2 c. f (x) = 7x2 d. f (x) = 23x + 6
The amount of spending on antipsychotic drugs, used to treat schizophrenia and other conditions, recently edged out cholesterol medications at the top of U.S. sales charts. The function a(x) = 1.24x
Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin. a. 3x2 + 4y2 = 5 b. y3 = 6x2
Determine no whether the function is even, odd, or neither even or odd. a. f (x) = 2x3 - x b. f (x) = 4x2 + 2x - 3
For each equation in Exercises, under the given condition: (a) Find k and (b) find a second solution. 1. kx2 - 17x + 33 = 0; one solution is 3 2. kx2 - 2x + k = 0; one solution is -3
In Exercises, use the given graph to find (a) the x-intercepts and (b) the zeros of the function.
Solve by completing the square to obtain exact solutions. a. x2 + 6x = 7 b. x2 + 8x = -15
Use the quadratic formula to find exact solutions. a. x2 - 2x = 15 b. x2 + 4x = 5 c. 5m2 + 3m = 2
For each of the following, find the discriminant, b2 - 4ac, and then determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions
Solve graphically. Round solutions to three decimal places, where appropriatea. x2 - 8x + 12 = 0b. 5x2 + 42x + 16 = 0
Find the zeros of the function algebraically. Give exact answers. a. f (x) = x2 + 6x + 5 b. f (x) = x2 - x - 2 c. f (x) = x2 - 3x - 3
Use a graphing calculator to find the zeros of the function. Round to three decimal placesa. f (x) = 3x2 + 2x - 4b. f (x) = 9x2 - 8x - 7
In Exercises, use the given graph to find (a) the vertex; (b) the axis of symmetry; and (c) the maximum or minimum value of the function.1.2.
In Exercises, match the equation with one of the graphs (a)-(h), which follow.1. y = (x + 3)2 2. y = -(x - 4)2 + 3 3. y = 2(x - 4)2 - 1 4. y = x2 - 3 5. y = - ½(x + 3)2 + 4
In Exercises, (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or minimum value and find that value; and (d) graph the function. 1. f (x) = x2 - 8x + 12 2.
In Exercises:a) Find the vertex.b) Determine whether there is a maximum or minimum value and find that value.c) Find the range.d) Find the intervals on which the function is increasing and the
A ball is thrown directly upward from a height of 6 ft with an initial velocity of 20 ft/sec. The function s(t) = -16t2 + 20t + 6 gives the height of the ball, in feet, t seconds after it has been
A stone is thrown directly upward from a height of 30 ft with an initial velocity of 60 ft/sec. The height of the stone, in feet, t seconds after it has been thrown is given by the function s(t) =
A model rocket is launched with an initial velocity of 120 ft/sec from a height of 80 ft. The height of the rocket, in feet, t seconds after it has been launched is given by the function s(t) = -16t2
A model rocket is launched with an initial velocity of 150 ft/sec from a height of 40 ft. The function s(t) = -16t2 + 150t + 40 gives the height of the rocket, in feet, t seconds after it has been
Mendoza Manufacturing plans to produce a one-compartment vertical file by bending the long side of a 10-in. by 18-in. sheet of plastic along two lines to form a U-shape. How tall should the file be
A fourth-grade class decides to enclose a rectangular garden, using the side of the school as one side of the rectangle. What is the maximum area that the class can enclose using 32 ft offence? What
The sum of the base and the height of a triangle is 20 cm. Find the dimensions for which the area is a maximum.
The sum of the base and the height of a parallelogram is 69 cm. Find the dimensions for which the area is a maximum.
Classic Furniture Concepts has determined that when x hundred wooden chairs are built, the average cost per chair is given byC(x) = 0.1x2 - 0.7x + 1.625,Where C(x) is in hundreds of dollars how many
In business, profit is the difference between revenue and cost; that is, Total profit = Total revenue - Total cost, P(x) = R(x) - C(x), Where x is the number of units sold. Find the maximum profit
A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If a river forms one side of the corrals and 240 yd of fencing is available, what is the largest total
A Norman window is a rectangle with a semicircle on top. Sky Blue Windows is designing a Norman window that will require 24 ft of trim on the outer edges. What dimensions will allow the maximum
Jenelle drops a screwdriver from the top of an elevator shaft. Exactly 5 sec later, she hears the sound of the screwdriver hitting the bottom of the shaft. How tall is the elevator shaft?
A water balloon is dropped from a cliff. Exactly 3 sec later, the sound of the balloon hitting the ground reaches the top of the cliff. How high is the cliff?
For each function f, construct and simplify the difference quotient f (x + h) - f (x) / h. a. f (x) = 3x - 7 b. f (x) = 2x2 - x + 4
A graph of y = f (x) follows. No formula is given for f. Make a hand-drawn graph of each of the following.a. g(x) = -2f (x) b. g(x) = f (2x)
Find c such that f (x) = -0.2x2 - 3x + c has a maximum value of -225.
Find b such that f (x) = -4x2 + bx + 3 has a maximum value of 50.
Find a quadratic function with vertex (4, -5) and containing the point (-3, 1).

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