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calculus

Questions and Answers of Calculus

The graphs labeled L1, L2, L3, and L4 represent four different pricing discounts, where p is the original price (in dollars) and S is the sale price (in dollars). Match each function with its graph.
(a) Given a function f, prove that g is even and h is odd, where g(x) = 1/2 [f (x) + f(ˆ’ x)] and h(x) = 1/2 [f (x) ˆ’ f (ˆ’x)].(b) Use the result of part (a) to prove
1. If f(g(x)) and g(f(x)) both equal x, then the function g is the ________ function of the function f. 2. The inverse function of f is denoted by ________. 3. The domain of f is the ________ of
The cost C for a business to make personalized T-shirts is given byC(x) = 7.50x + 1500where x represents the number of T-shirts.(a) The graphs of C and Cˆ’1 are shown below. Match each
1. The number of miles n a marathon runner has completed in terms of the time t in hours 2. The depth of the tide d at a beach in terms of the time t over a 24-hour period determine whether the
Verify that f and g are inverse functions algebraically. 1. f(x) = x - 9/4, g(x) = 4x + 9 2. f(x) = -3/2x - 4, g(x) = - 2x + 8/3
Use the graph of the function to sketch the graph of its inverse function y = f-1(x).1.2.
Verify that f and g are inverse functions (a) Algebraically (b) Graphically. 1. f(x) = x − 5, g(x) = x + 5 2. f(x) = 7x + 1, g(x) = x - 1/7
Does the function have an inverse function?1.2.
Use the table of values for y = f(x) to complete a table for y = f-1 (x).1.2.
Does the function have an inverse function?1.2. 3. 4.
Use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. 1. g(x) = (x + 3)2 + 2 2. f(x) = 1/5(x + 2)3 3. f(x) = x √9
(a) Find the inverse function of f, (b) Graph both f and f-1 on the same set of coordinate axes, (c) Describe the relationship between the graphs of f and f-1, (d) State the domains and ranges of f
Determine whether the function has an inverse function. If it does, find the inverse function. 1. f(x) = x4 2. f(x) = 1/x2 3. g(x) = x + 1/6
Find the inverse function of f informally. Verify that f (f-1 (x)) = x and f-1(f (x)) = x. 1. f(x) = 6x 2. f(x) = 1/3x 3. f(x) = 3x + 1
Restrict the domain of the function f so that the function is one-to-one and has an inverse function. Then find the inverse function f-1. State the domains and ranges of f and f-1. Explain your
Use the functions f(x) = 1/8 x - 3 and g(x) = x3 to find the value or function. 1. (f-1 ∘ g-1) (1) 2. (g-1 ∘ f-1) (-3)
Use the functions f(x) = x + 4 and g(x) = 2x - 5 to find the function. 1. g-1 ∘ f-1 2. f-1 ∘ g-1
Your wage is $10.00 per hour plus $0.75 for each unit produced per hour. So, your hourly wage y in terms of the number of units produced x is y = 10 + 0.75x. (a) Find the inverse function. What does
The functiony = 0.03x2 + 245.50, 0 approximates the exhaust temperature y in degrees Fahrenheit, where x is the percent load for a diesel engine.(a) Find the inverse function. What does each variable
Determine whether the statement is true or false. Justify your answer. 1. If f is an even function, then f−1 exists. 2. If the inverse function of f exists and the graph of f has a y-intercept,
Use the graph of the function f to create a table of values for the given points. Then create a second table that can be used to find f-1, and sketch the graph of f-1, if possible.1.2.
Prove that if f and g are one-to-one functions, then (f ∘ g)−1(x) = (g−1 ∘ f −1)(x).
Prove that if f is a one-to-one odd function, then f−1 is an odd function.
The function f(x) = k(2 − x − x3) has an inverse function, and f−1(3) = −2. Find k.
Consider the functions f (x) = x + 2 and fˆ’1(x) = x ˆ’ 2. Evaluate f (fˆ’1(x)) and fˆ’1(f (x)) for the given values of x. What can you conclude about the
Restrict the domain of f (x) = x2 + 1 to x ≥ 0. Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.
Find equations of the lines that pass through the given point and are (a) Parallel to (b) Perpendicular to the given line. 5x − 4y = 8, (3, −2)
A discount outlet offers a 20% discount on all items. Write a linear equation giving the sale price S for an item with a list price L.
A manuscript translator charges a starting fee of $50 plus $2.50 per page translated. Write a linear equation for the amount A earned for translating p pages.
Determine whether the equation represents y as a function of x. 1. 16x − y4 = 0 2. 2x − y − 3 = 0 3. y = √1 - x
Find each function value. 1. g(x) = x4/3 (a) g(8) (b) g(t + 1) (c) g(−27) (d) g(−x) 2. h(x) = |x - 2| (a) h(−4) (b) h(−2) (c) h(0) (d) h(−x + 2)
Find the domain of the function. 1. f(x) = √25 - x2 2. h(x) = x/x2 - x - 6
The velocity of a ball projected upward from ground level is given by v(t) = -32t + 48, where t is the time in seconds and v is the velocity in feet per second. 1. Find the velocity when t = 1. 2.
Find the difference quotient and simplify your answer. f (x) = 2x2 + 3x − 1, f(x + h)/h, h ≠ 0
Use the Vertical Line Test to determine whether the graph represents y as a function of x. To print an enlarged copy of the graph, go to MathGraphs.com.1.2.
Find the zeros of the function algebraically. 1. f(x) = 5x2 + 4x - 1 2. f(x) = 8x + 3/11 - x
Use a graphing utility to graph the function and visually determine the open intervals on which the function is increasing, decreasing, or constant. 1. f (x) = |x| + |x + 1| 2. f (x) = (x2 − 4)2
Use a graphing utility to approximate (to two decimal places) any relative minima or maxima of the function. 1. f (x) = −x2 + 2x + 1 2. f (x) = x3 − 4x2 - 1
Find the average rate of change of the function from x1 to x2. 1. f (x) = −x2 + 8x − 4, x1 = 0, x2 = 4 2. f (x) = x3 + 2x + 1, x1 = 1, x2 = 3
Determine whether the function is even, odd, or neither. Then describe the symmetry. 1. f (x) = x5 + 4x - 7 2. f (x) = x4 − 20x2
(a) write the linear function f that has the given function values (b) sketch the graph of the function. f(2) = −6, f(−1) = 3
Sketch the graph of the function.1. g(x) = [x] ˆ’ 22. g(x) = [x + 4]3.4.
h is related to one of the parent functions described in this chapter. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to h. (c) Sketch the graph of h. (d) Use
Find the slope of the line passing through the pair of points. 1. (5, −2), (−1, 4) 2. (−1, 6), (3, −2)
Find (a) (f + g) (x), (b) (f - g) (x), (c) ( fg)(x), (d) (f/g)(x). What is the domain of f/g? 1. f (x) = x2 + 3, g(x) = 2x − 1 2. f (x) = x2 − 4, g(x) = √3 − x
Find (a) f ∘ g (b) g ∘ f. Find the domain of each function and of each composite function. f (x) = 1/3x − 3, g(x) = 3x + 1
The price of a washing machine is x dollars. The function f(x) = x - 100 gives the price of the washing machine after a $100 rebate. The function g(x) = 0.95x gives the price of the washing machine
Find the inverse function of f informally. Verify that f(f-1(x)) = x and f-1(f(x)) = x. 1. f(x) = x - 4/5 2. f(x) = x3 - 1
Use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. 1. F(x) = (x - 1)2 2. h(t) = 2/t - 3
(a) Find the inverse function of f, (b) Graph both f and f-1 on the same set of coordinate axes, (c) Describe the relationship between the graphs of f and f-1, (d) State the domains and ranges of f
Find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line. 1. m = 1/3, (6, −5) 2. m = −3/4, (−4, −2)
Restrict the domain of the function f to an interval on which the function is increasing, and find f-1 on that interval. 1. f (x) = 2(x − 4)2 2. f (x) = |x - 2|
Relative to the graph of f (x) = √x, the graph of the function h(x) = −√x + 9 − 13 is shifted 9 units to the left and 13 units down, then reflected in the x-axis. Determine whether the
If f and g are two inverse functions, then the domain of g is equal to the range of f. Determine whether the statement is true or false. Justify your answer.
Find an equation of the line passing through the pair of points. Sketch the line. 1. (−6, 4), (4, 9) 2. (−9, −3), (−3, −5)
Find an equation of the line passing through the pair of points. Sketch the line. 1. (−2, 5), (1, −7) 2. (−4, −7), (1, 4/3)
Use a graphing utility to approximate (to two decimal places) any relative minima or maxima of f (x) = −x3 + 2x − 1.
Find the average rate of change of f (x) = −2x2 + 5x − 3 from x1 = 1 to x2 = 3.
(a) identify the parent function f in the transformation,€–(b)describe the sequence of transformations from f to h.(c)sketch the graph of h.2. h (x) = - ˆš(x + 5) + 8 3. h(x) = -2
Find, (a) (f + g) (x), (b) (f - g) (x), (c) (fg) (x), (d) (f/g) (x), (e) (fog) (x), (f) (gof) (x) 1. f (x) = 3x2 - 7, g (x) = -x2 -4x + 5 2. f (x) = 1 / x, g (x) = 2 √x
Determine whether the function has an inverse function. If it does, find the inverse function. 1. f (x) = x2 + 8 2. f (x) = |x2 - 3| + 6 3. f (x) = 3x √x
It costs a company $58 to produce 6 units of a product and $78 to produce 10units. Assuming that the cost function is linear, how much does it cost to produce 25units?
Find equations of the lines that pass through the point (0, 4) and are (a) parallel to and (b) perpendicular to the line 5x + 2y = 3.
Find each function value. 1. f (x) = |x + 2| - 15, (a) f (-8) (b) f (14) (c) f (x - 6) 2. f (x) = √(x + 9) / (x2 - 81) (a) f (7) (b) f (-5) (c) f (x - 9)
Find (a) the domain and (b) the zeroes of the function. 1. f (x) = (x -5) / (2x2 - x) 2. f (x) = 10 - √(3 - x)
(a) use a graphing utility to graph the function, (b) approximate the open intervals on which the function is increasing, decreasing, or constant, (c) Determine whether the function is even, odd,
1. Linear, constant, and squaring functions are examples of ________ functions. 2. A polynomial function of x with degree n has the form f (x) = anxn + an−1xn 1 + . . . + a1x + a0 (an ≠ 0), where
In Exercises 1-2, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and x-intercept(s). 1. f(x) = x2 - 6x 2. g(x) = x2 - 8x
In Exercises 1-4, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercept(s). Then check your results algebraically by writing the quadratic
In Exercises 1-36, write the standard form of the quadratic function whose graph is the parabola shown.1.2.
In Exercises 1-4, write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point. 1. Vertex: (−2, 5); point: (0, 9) 2.
In Exercises 1-2, determine the x-intercept(s) of the graph visually. Then find the x-intercept(s) algebraically to confirm your results.1. y = x2 ˆ’ 2x - 32. y = x2 ˆ’ 4x - 5 3.
In Exercises 1-4, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), and (d).]a.b.c.d.1. f (x) = x2 âˆ’ 22. f (x) = (x + 1)2 âˆ’ 23. f (x) = âˆ’(x âˆ’
In Exercises 1-2, use a graphing utility to graph the quadratic function. Find the x-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when f(x) =
In Exercises 1-2, find two quadratic functions, one that opens upward and one that opens downward; whose graphs have the given x-intercepts. (There are many correct answers.) 1. (−3, 0), (3, 0) 2.
In Exercises 1-2, find two positive real numbers whose product is a maximum. 1. The sum is 110. 2. The sum is S.
The path of a diver is modeled by f(x) = - 4/9 x2 + 24/9 x + 12 where f (x) is the height (in feet) and x is the horizontal distance (in feet) from the end of the diving board. What is the maximum
Height of a Ball The path of a punted football is modeled by f(x) = -16 / 2025 x2 + 9/5 x + 1.5 where f (x) is the height (in feet) and x is the horizontal distance (in feet) from the point at which
A manufacturer of lighting fixtures has daily production costs of C = 800 − 10x + 0.25x2, where C is the total cost (in dollars) and x is the number of units produced. What daily production number
The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model P = 230 + 20x − 0.5x2. What
The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by R(p) = −25p2 + 1200p, where p is the price per unit (in dollars). (a) Find the revenues when
Maximum Revenue The total revenue R earned per day (in dollars) from a pet-sitting service is given by R(p) = −12p2 + 150p, where p is the price charged per pet (in dollars). (a) Find the revenues
A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).(a) Write the area A of the corrals as a function of x. (b) What dimensions produce a maximum enclosed area?
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 16 feet.(a) Write the area A of the window as a
In Exercises 1 and 2, determine whether the statement is true or false. Justify your answer. 1. The graph of f (x) = −12x2 − 1 has no x-intercepts. 2. The graphs of f (x) = −4x2 − 10x + 7 and
In Exercises 1 and 2, find the values of b such that the function has the given maximum or minimum value. 1. f (x) = −x2 + bx − 75; Maximum value: 25 2. f (x) = x2 + bx − 25; Minimum value:
Write the quadratic function f (x) = ax2 + bx + c in standard form to verify that the vertex occurs at ( -b/2a, f(-b/2a)).
Assume that the function f (x) = ax2 + bx + c, a ≠ 0 Have two real zeros. Prove that the x-coordinate of the vertex of the graph is the average of the zeros of f.
In Exercises 1-2, sketch the graph of each quadratic function and compare it with the graph of y = x2. 1. a. f(x) = 1/2x2 b. g(x) = -1/8x2 c. h(x) = 3/2x2 d. k(x) = -3x2 2. a. f(x) = x2 +1 b. g(x) =
1. The graph of a polynomial function is ________, which means that the graph has no breaks, holes, or gaps. 2. The ________ ________ ________ is used to determine the left-hand and right-hand
Sketch the graph of a fourth-degree polynomial function that has a zero of multiplicity 2 and a negative leading coefficient. Sketch the graph of another polynomial function with the same
Sketch the graph of a fifth-degree polynomial function that has a zero of multiplicity 2 and a negative leading coefficient. Sketch the graph of another polynomial function with the same
Sketch the graph of the function f (x) = x4. Explain how the graph of each function g differs (if it does) from the graph of f. Determine whether g is even, odd, or neither. (a) g(x) = f (x) + 2 (b)
Use a graphing utility to graph the functions y1 = -1/3 (x - 2)5 + 1 and y2 = 3/5 (x + 2)5 - 3. a. Determine whether the graphs of y1 and y2 are increasing or decreasing. Explain. b. Will the graph
In Exercises 1-2, sketch the graph of y = xn and each transformation. 1. y = x3 a. f(x) = (x - 4)3 b. f(x) = x3 - 4 c. f(x) = 1-1/4x3 d. f(x) = (x - 4)3 - 4 2. y = x5 a. f(x) = (x + 1)5 b. f(x) = x5
In Exercises 1-4, describe the left-hand and right-hand behavior of the graph of the polynomial function. 1. f(x) = 12x3 + 4x 2. f(x) = 2x2 - 3x + 1 3. g(x) = 5 - 1/2x - 3x2 4. h(x) = 1 - x6
In Exercises 1-2, use a graphing utility to graph the functions f and g in the same viewing window. Zoom out sufficiently far to show that the left-hand and right-hand behaviors of f and g appear
In Exercises 1-2, (a) find all real zeros of the polynomial function, (b) determine whether the multiplicity of each zero is even or odd, (c) determine the maximum possible number of turning
In Exercises 1-2, (a) use a graphing utility to graph the function, (b) use the graph to approximate any x-intercepts of the graph, (c)find any real zeros of the function algebraically,

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