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linear algebra
Questions and Answers of
Linear Algebra
If [x + 2] = -3, what are the possible inputs for x?
If ([x +])2 = 25, what are the possible inputs for x?
A power line is constructed from a power station at point A to an island at point I, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi downshore from the power
A right circular cylinder of height h and radius r is inscribed in a right circular cone with a height of 10 ft and a base with radius 6 ft.(a) Express the height h of the cylinder as a function of
Given that f(x) = x2 - 3 and g(x) = 2x + 1, find each of the following, if it exists. (a) (f + g) (5) (b) (fg) (0) (c) (f - g) (- 1)
Given that h(x) = x + 4 and g(x) = √x - 1, find each of the following, if it exists. (a) (h - g) (- 4) (b) (gh) (10) (c) (g / h) (1)
For each pair of function in Exercise f(x) = 2x + 3, g(x) = 3 - 5x (a) Find the domain of f, g, f + g, f - g, fg, f f, f/g, and g/f. (b) Find (f + g) (x), (f- g) (x), (fg) (x), (ff) (x), (f/g) (x),
Consider the functions F and G as shown in the graph below.(a) Find the domain of F, the domain of G, and the domain of F + G. (b) Find the domain of F - G, FG, and F/G. (c) Find the domain of G/F.
Consider the functions F and G as shown in the graph below.(a) Find the domain of F, the domain of G, and the domain of F + G. (b) Find the domain of F - G, FG, and F/G. (c) Find the domain of G/F.
In economics, functions that involve revenue, cost, and profit are used. For example, suppose that R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new kind of
Given that R(x) = 200x - x2 and C(x) = 5000 + 8x for a new weather radio produced by Clear Communication, find each of the following. (a) P(x) (b) R(175), C(175), and P(175)
For each function f, construct and simplify the difference quotient f(x + h) - f(x) / h. (a) f(x) = 3x - 5 (b) f(x) = 4x - 1 (c) f(x) = 6x + 2
Graph the equation. (a) y = 3x - 1 (b) 2x + y = 4 (c) x - 3y = 3
For functions h and f, find the domain of h + f, h - f, hf, and h/f if h = {(-4, 13), (- 1, 7), (0, 5), (5/2, 0), (3, -5), and f = {(-4, -7), (-2, -5), (0, -3), (3, 0), (5, 0), (9, 6)}.
Find the domain of (h/g) given that h(x) = 5x / 3x - 7 and g(x) = x4 - 1 / 5x - 15.
Given that f(x) = 3x + 1, g(x) = x2 - 2x - 6, and h(x) = x3, find each of the following. (a) (fog) (- 1) (b) (fof) (- 2) (c) (hof) (1)
Find 1f (fg) and (gf) (x) and the dmain f each.(a) f(x) = x + 3, g(x) = x - 3(b) f(x) = 4/5 x, g(x) = 5/4 x(c) f(x) = x + 1, g(x) = 3x2 - 2x - 1
Find f(x) and g(x) such that h(x) = 1fog(x). (a) h(x) = (4 + 3x)5 (b) h(x) = 3√x2 - 8 (c) h(x) = 1 / (x - 2)4
A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3 ft/sec.(a) Find a function r(t) for the radius in terms
The surface area S of a right circular cylinder is given by the formula S = 2prh + 2pr2.If the height is twice the radius, find each of the following.(a) A function S1r2 for the surface area as a
A manufacturer of tools, selling rechargeable drills to a chain of home improvement stores, charges $2 more per drill than its manufacturing cost m. The chain then sells each drill for 150% of the
A blouse that is size x in Japan is size s(x) in the United States, where s(x) = x - 3. A blouse that is size x in the United States is size t(x) in Australia, where t(x) = x + 4. Find a function
Consider the following linear equations. Without graphing them, answer the questions below. (a) y = x (b) y = 2 / 3 x + 1 (c) y = -0.1x + 6 (a) Which, if any, have y-intercept (0, 1)? (b) Which slope
Determine visually whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin.(a)(b) (c)
Test algebraically whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. 5x - 5y = 0
Find the point that is symmetric to the given point with respect to the x-axis, the y-axis, and the origin. (a) (-5, 6) (b) (7/2, 0) (c) (-10, -7)
Determine visually whether the function is even, odd,or neither even nor odd.(a)(b) (c)
Determine algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator. (a) f(x) = -3x3 + 2x (b) f(x) = 7x3
Determine whether the function is even, odd, or neither even nor odd. (a) f(x) = x√10 - x2 (b) f(x) = x2 + / x3 - 1
Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin.(a) y2 + 4xy2 - y4 = x4 - 4x3 + 3x2 +2x2y2(b) (x2 + y2)2 = 2xy
Show that if f is any function, then the function O defined by O(x) = f(x) - f(-x) / 2 is odd.
State whether each of the following is true or false. (a) The product of two odd functions is odd. (b) The sum of two even functions is even. (c) The product of an even function and an odd function
First, graph the 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) y = | x | = - 2
Describe how the graph of the function can be obtained from one of the basic graphs on p. 199. Then graph the function by hand or with a graphing calculator. (a) f(x) = (x - 3)2 (b) g(x) = x2 +
Describe how the graph of the function can be obtained from one of the basic graphs on p. 199. (a) g(x) = |3x| (b) f(x) = 1/2 3√x (c) h(x) = 2/x
The point (-12, 4) is on the graph of y = f(x). Find the corresponding point on the graph of y = g(x). (a) g(x) = 1/2 f(x) (b) g(x) = f(x - 2) (c) g(x) = f(-x)
Given that f(x) = x2 + 3, match the function g with a transformation of f from one of A-D. (a) g(x) = x2 + 4 f(x - 2) (b) g(x) = 9x2 + 3 f(x) + 1 (c) g(x) = (x - 2)2 + 3 2f(x) (d) g(x) = 2x2 + 6 f(3x)
The shape of y = x2, but reflected across the x-axis and shifted right 8 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = √x, but shifted left 6 units and down 5 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = | x |, but shifted left 7 units and up 2 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = x3, but reflected across the x-axis and shifted right 5 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = 1/x, but shrunk horizontally by a factor of 2 and shifted down 3 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = x2, but shifted right 6 units and up 2 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = x2, but reflected across the x-axis and shifted right 3 units and up 4 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = |x|, but stretched horizontally by a factor of 2 and shifted down 5 units Write an equation for a function that has a graph with the given characteristics.
The shape of y = √x, but reflected across the y-axis and shifted left 2 units and down 1 unit Write an equation for a function that has a graph with the given characteristics.
The shape of y = 1/x, but reflected across the x-axis and shifted up 1 unit Write an equation for a function that has a graph with the given characteristics.
A graph of y = f(x) follows. No formula for f is given. In Exercises graph the given equation.(a) G(x) = - 2f(x) (b) g(x) = 1/2 f(x) (c) g(x) = f(- 1/2 x)
A graph of y = g(x) follows. No formula for g is given. In Exercises graph the given equation.(a) h(x) = - g(x + 2) + 1 (b) h(x) = 1/2 g(- x) (c) h(x) = g(2x)
For each pair of functions, determine if g(x) = f(-x). (a) f(x) = 2x4 - 35x3 + 3x - 5, g(x) = 2x4 + 35x3 - 3x - 5 (b) f(x) = 1/4 x4 + 1/5 x3 - 81x2 - 17, g(x) = 1/4 x4 + 1/5 x3 + 81x2 - 17
A graph of the function f(x) = x3 - 3x2 is shown below. Exercises show graphs of functions transformed from this one. Find a formula for each function.(a) (b) (c)
Determine algebraically whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin. (a) y = 3x4 - 3 (b) y2 = x (c) 2x - 5y = 0
Sales of the video game Wii Fit totaled 3.5 million games in the first eleven months of 2009. This was 1 million less than three times the number of Madden NFL 10 games sold during the same period.
It is estimated that about $5 billion in gift cards given as Christmas gifts in 2009 went unspent. This is about 6% of the total amount spent on gift cards. Find the total amount spent on gift cards.
Use the graph of the function f shown below in Exercises.(a) Graph: y = |f (x)|. (b) Graph: y = f (|x|).
Use the graph of the function g shown below in Exercises(a) Graph: y = g(|x|). (b) Graph: y = |g(x)|.
Graph each of the following using a graphing calculator. Before doing so, describe how the graph can be obtained from a more basic graph. Give the domain and the range of the function. (a) f(x) = [x
If (3, 4) is a point on the graph of y = f(x), what point do you know is on the graph of y = 2f(x)? of y = 2 + f(x)? of y = f(2x)?
Find the zeros of f(x) = 3x5 - 20x3. Then, without using a graphing calculator, state the zeros of f(x - 3) and f(x + 8).
Find the variation constant and an equation of variation for the given situation. (a) y varies directly as x, and y = 54 when x = 12 (b) y varies directly as x, and y = 0.1 when x = 0.2 (c) y varies
The amount of sales tax paid on a product is directly proportional to its purchase price. In Indiana, the sales tax on a Sony Reader that sells for $260 is $17.50. What is the sales tax on an e-book
The Gemmers decide to give their children a weekly allowance that is directly proportional to each child's age. Their 6-year-old daughter receives an allowance of $4.50. What is their 11-year-old
The weight W that a horizontal beam can support varies inversely as the length L of the beam. Suppose an 8-m beam can support 1200 kg. How many kilograms can a 14-m beam support?
The time t required to drive a fixed distance varies inversely as the speed r. It takes 5 hr at a speed of 80 km/h to drive a fixed distance. How long will it take to drive the same distance at a
The maximum number of grams of fat that should be in a diet varies directly as a person's weight. A person weighing 120 lb should have no more than 60 g of fat per day. What is the maximum daily fat
The number of representatives N that each state has varies directly as the number of people P living in the state. If New York, with 19,254,630 residents, has 29 representatives, how many
The time T required to do a job varies inversely as the number of people P working. It takes 5 hr for 7 bricklayers to build a park wall. How long will it take 10 bricklayers to complete the job?
The time t required to empty a tank varies inversely as the rate r of pumping. If a pump can empty a tank in 45 min at the rate of 600 kL/min, how long will it take the pump to empty the same tank at
Hooke's law states that the distance d that a spring will stretch varies directly as the mass m of an object hanging from the spring. If a 3-kg mass stretches a spring 40 cm, how far will a 5-kg mass
The pitch P of a musical tone varies inversely as its wavelength W. One tone has a pitch of 330 vibrations per second and a wavelength of 3.2 ft. Find the wavelength of another tone that has a pitch
The weight M of an object on Mars varies directly as its weight E on Earth. A person who weighs 95 lb on Earth weighs 38 lb on Mars. How much would a 100-lb person weigh on Mars?
y varies inversely as the square of x, and y = 0.15 when x = 0.1 Find an equation of variation for the given situation.
y varies directly as the square of x, and y = 0.15 when x = 0.1 Find an equation of variation for the given situation.
y varies jointly as x and z, and y = 56 when x = 7 and z = 8 Find an equation of variation for the given situation.
y varies jointly as x and the square of z, and y = 105 when x = 14 and z = 5 Find an equation of variation for the given situation.
y varies jointly as x and z and inversely as the product of w and p, and y = 3 / 28 when x = 3, z = 10, w = 7, and p = 8 Find an equation of variation for the given situation.
y varies jointly as x and z and inversely as the square of w, and y = 12 / 5 when x = 16, z = 3, and w = 5 Find an equation of variation for the given situation.
The intensity I of light from a light bulb varies inversely as the square of the distance d from the bulb. Suppose that I is 90 W/m2 (watts per square meter) when the distance is 5 m. How much
Wind resistance, or atmospheric drag, tends to slow down moving objects. Atmospheric drag varies jointly as an object's surface area A and velocity v. If a car traveling at a speed of 40 mph with a
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 fast can a car travel and still
The weight W of an object varies inversely as the square of the distance d from the center of the earth. At sea level (3978 mi from the center of the earth), an astronaut weighs 220 lb. Find his
A pitcher's earned-run average E varies directly as the number R of earned runs allowed and inversely as the number I of innings pitched. In 2010, Bronson Arroyo of the Cincinnati Reds had an
The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V = 231 cm3 when T = 42o and P = 20 kg/cm2, what is the volume when T = 30o and P = 15
In each of the following equations, state whether y varies directly as x, inversely as x, or neither directly nor inversely as x. (a) 7xy = 14 (b) x - 2y = 12 (c) -2x + 3y = 0 (d) x = 3 / 4 y (e) x /
An 18-oz jar of peanut butter in the shape of a right circular cylinder is 5 in. high and 3 in. in diameter and sells for $2.89.In the same store, a 28-oz jar of the same brand is 5 1/2 in. high and
The area of a circle varies directly as the square of the length of a diameter. What is the variation constant?
Given that f(x) = 3x - 1 and g(x) = x2 + 4, find each of the following, if it exists.(a) (f + g) (-1)(b) (fg) (0)(c) (g - f) (3)(d) (g/f) (1/3)
For each pair of functions in Exercise f(x) = 2x + 5; g(x) = - x - 4 (a) Find the domains of f, g, f + g, f - g, fg, f f, f/g, and g/f. (b) Find (f + g) (x), (f - g) (x), (fg) (x), (ff), (x), (f/g)
For each function f, construct and simplify the difference quotient f(x + h) - f(x) / h. (a) f(x) = 4x - 3 (b) f(x) = 6 - x2
Given that f(x) = 5x - 4, g(x) = x3 + 1, and h(x) = x2 - 2x + 3, find each of the following. (a) (fog) (1) (b) (goh) (2) (c) (fof) (0) (d) (hof) (-1)
Find (fog) and (gof)(x) and the domain of each. (a) f(x) = 1/2 x, g(x) = 6x + 4 (b) f(x) = 3x + 2, g(x) = √x
If f and g are linear functions, what can you say about the domain of fog and the domain of gof?
Determine the intervals on which the function is (a) increasing; (b) decreasing; (c) constant.
Using the graph, determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing.
For the function defined asfind f(-5), f(-3), f(-1), and f(6).
Graph the function defined as
Determine whether the statement is true or false. (a) The greatest integer function pairs each input with the greatest integer less than or equal to that input. (b) In general, for functions f and g,
A seamstress uses 20 ft of lace to trim the edges of a rectangular tablecloth. If the tablecloth is l feet long, express its area as a function of the length.
A rectangle is inscribed in a semicircle of radius 2, as shown. The variable x = half the length of the rectangle. Express the area of the rectangle as a function of x.
Mamie has 66 ft of fencing with which to enclose a rectangular dog pen. The side of her garage forms one side of the pen. Suppose the side of the pen parallel to the garage is x feet long. (a)
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