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Question: Evaluate the difference quotient for the given

Evaluate the difference quotient for the given function. Simplify your answer.
Evaluate the difference quotient for the given function.  Simplify your answer.





Transcribed Image Text:

x + 3 f(x) = x + 1' f(x) – f(1) x - 1


> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are functions, then f 0 g = g 0 f.

> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A vertical line intersects the graph of a function at most once.

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> Sketch the graph of the function f(x) = |x2 - 4|x|+ 3|.

> Solve the inequality |x - 1|-|x - 3|≥ 5.

> Find the domain of the function. g(1) = /3 – t – /2 + t 3 -

> Solve the equation |2x - 1|-|x + 5|= 3.

> The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.

> One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.

> a. If f0(x) = 1 / 2 - x and fn+1 = f0 0 fn for n = 0, 1, 2, ..., find an expression for fn(x) and use mathematical induction to prove it. b. Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition

> If f0(x) = x2 and fn+1(x) − f0(fn(x)) for n = 0, 1, 2, ..., find a formula for fn(x).

> Prove that 1 + 3 + 5 + ∙∙∙ + (2n - 1) = n2.

> Prove that if n is a positive integer, then 7n - 1 is divisible by 6.

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> A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 mi/h; she drives the second half at 60 mi/h. What is her average speed on this trip?

> Use indirect reasoning to prove that log25 is an irrational number.

> Find the domain of the function. f(1) = /2t – 1

> Solve the inequality ln(x2 - 2x – 2) ≤ 0.

> a. Show that the function f(x) = ln(x + x2 + 1 ) is an odd function. b. Find the inverse function of f.

> Evaluate (log23)(log34)(log45)∙∙∙(log3132).

> Sketch the region in the plane defined by each of the following equations or inequalities. a. max{x, 2y} = 1 b. -1 ≤ max{x, 2y} ≤ 1 c. max{x, y2} = 1

> The notation max{a, b, ...} means the largest of the numbers a, b, ... . Sketch the graph of each function. a. f(x) = max{x, 1/x} b. f(x) = max{sin x, cos x} c. f(x) = max{x2, 2 + x, 2 - x}

> Sketch the region in the plane consisting of all points (x, y) such that |x - y|+|x|-|y|≤ 2

> Draw the graph of the equation x +|x|= y +|y|.

> Sketch the graph of the function g(x) =|x2 - 1|-|x2 - 4|.

> If f(x) = x2 - 2x + 3, evaluate the difference quotient f (a+h) – f (a) / h

> The graph of g is given. a. State the value of g(2). b. Why is g one-to-one? c. Estimate the value of g-1(2). d. Estimate the domain of g-1. e. Sketch the graph of g-1. 0 1

> Find the domain of the function. 2x – 5 - 5 .3 f(x) x? + x – 6

> Let f be the function whose graph is given. a. Estimate the value of f(2). b. Estimate the values of x such that f(x) = 3. c. State the domain of f. d. State the range of f. e. On what interval is f increasing? f. Is f one-to-one? Explain. g. Is f

> The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is P(t) = 100,000/100 + 900e-t where t is measured in years. a. Graph this function and estimate how long it takes for the population to

> The half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days.) The initial mass of a sample is one gram. a. Find the mass that remains after 16 days. b. Find the mass m(t) that remains afte

> Solve each equation for x. a. ex = 5 b. lnx = 2 c. eex = 2 d. tan-1x = 1

> Find the exact value of each expression. a. e2ln3 b. log1025 + log104 c. tan(arcsin ½) d. sin(cos-1 (4/5))

> Find the inverse function of f(x) = x + 1 / 2x + 1.

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> Use transformations to sketch the graph of the function. y = (x - 2)3

> The graph of f is given. Draw the graphs of the following functions. a. y = f(x &acirc;&#128;&#147; 8) b. y = -f(x) c. y = 2 &acirc;&#128;&#147; f(x) d. y = 1/2 f(x) - 1 e. y = f -1(x) f. y = f -1(x + 3) yA 1 1

> The graph of a function f is given. a. State the value of f(1). b. Estimate the value of f(-1). c. For what values of x is f(x) &acirc;&#136;&#146; 1? d. Estimate the value of x such that f(x) &acirc;&#136;&#146; 0. e. State the domain and range of f

> Suppose that the graph of f is given. Describe how the graphs of the following functions can be obtained from the graph of f. a. y = f(x) + 8 b. y = f(x + 8) c. y = 1 + 2f(x) d. y = f(x – 2) - 2 e. y = -f(x) f. y = f -1(x)

> Find the domain and range of the function. Write your answer in interval notation. f(t) = 3 + cos 2t

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> Find the domain and range of the function. Write your answer in interval notation. f(x) = 2/(3x – 1)

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> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. 1 3 4 5 f(x) 1.5 2.0 3.6 5.3 2.8 2.0

> a. Suppose f is a one-to-one function with domain A and range B. How is the inverse function f-1 defined? What is the domain of f -1? What is the range of f -1? b. If you are given a formula for f, how do you find a formula for f -1? c. If you are give

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> Evaluate the difference quotient for the given function. Simplify your answer. 1 f(x) f(x) – f(a) х — а

> a. If we shift a curve to the left, what happens to its reflection about the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c), where f is a one-to-one function. b. Find an expression for the inverse

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> Graph the given functions on the same screen. How are these graphs related? y = sin x, -π/2 ≤ x ≤ π/2; y = sin-1x; y = x

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> Find the exact value of each expression. a. csc-1 √2 b. arcsin 1

> Find the exact value of each expression. a. tan-1 3 b. arctan(-1)

> Find the exact value of each expression. a. cos-1(-1) b. sin-1(0.5)

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> Graph the function f(x) = x3 + x2 + x + 1 and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for f -1(x). (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this

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> Solve each inequality for x. a. ln x < 0 b. ex > 5

> Solve each equation for x. a. ln(ln x) = 1 b. eax = Cebx, where a ≠ b

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> Solve each equation for x. a. ln(x2 – 1) − 3 b. e2x - 3ex + 2 = 0

> Solve each equation for x. a. e7-4x = 6 b. ln(3x - 10) = 2

> a. What are the domain and range off? b. What is the x-intercept of the graph off? c. Sketch the graph off. f(x) = ln(x – 1) - 1

> a. What are the domain and range off? b. What is the x-intercept of the graph off? c. Sketch the graph off. f(x) = ln x + 2

> Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. a. y = ln (-x) b. y = ln|x| From Figures 12 and 13 y= log, x y= log

> A spherical balloon with radius r inches has volume V(r) = 4/3πr3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches.

> Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. a. y = log10(x + 5) b. y = -ln x From Figures 12 and 13 y= log, x y

> Compare the functions f(x) = x0.1 and g(x) = ln x by graphing both f and g in several viewing rectangles. When does the graph off finally surpass the graph of g?

> Suppose that the graph of y = log2x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?

> Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y = ln x, y = log10x, y = ex, y = 10x

> Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y = log1.5x, y = ln x, y = log10x, y = log50x

> Use Formula 10 to evaluate each logarithm correct to six decimal places. a. log510 b. log357

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