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 given the graph of f, how do you find the graph of f -1?
> 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.
> Is it true that f 0 (g + h) = f 0 g + f 0 h?
> 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.
> If f(x) = 2x + lnx, find f -1(2).
> A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. a. Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then s
> Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a
> Express the function f(x) = 1/ x + x as a composition of three functions.
> Find the domain of the function. x + 4 f(x) x? – 9
> If f(x) = lnx and g(x) = x2 - 9, find the functions a. f 0 g, b. g 0 f, c. f 0 f, d. g 0 g, and their domains.
> Find an expression for the function whose graph consists of the line segment from the point (-2, 2) to the point (-1, 0) together with the top half of the circle with center the origin and radius 1.
> Determine whether f is even, odd, or neither even nor odd. a. f(x) = 2x5 - 3x2 + 2 b. f(x) = x3 - x7 c. f(x) = e-x2 d. f(x) = 1 + sinx
> Use transformations to sketch the graph of the function. if x< 0 f(x) = et – 1 if x> 0
> Use transformations to sketch the graph of the function. f(x) = -cos 2x
> Use transformations to sketch the graph of the function. y = ln(x + 1)
> Use transformations to sketch the graph of the function. y = x2 - 2x + 2
> Use transformations to sketch the graph of the function. y = 2
> 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 – 8) b. y = -f(x) c. y = 2 – f(x) d. y = 1/2 f(x) - 1 e. y = f -1(x) f. y = f -1(x + 3) yA 1 1
> Evaluate the difference quotient for the given function. Simplify your answer. x + 3 f(x) = x + 1' f(x) – f(1) x - 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) − 1? d. Estimate the value of x such that f(x) − 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
> Find the domain and range of the function. Write your answer in interval notation. h(x) = ln(x + 6)
> Find the domain and range of the function. Write your answer in interval notation. g(x) = 16 −
> Find the domain and range of the function. Write your answer in interval notation. f(x) = 2/(3x – 1)
> Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used.
> 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 2 3 4 5 6. f(x) 1.0 1.9 2.8 3.5 3.1 2.9
> 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. What is a one-to-one function? b. How can you tell from the graph of a function whether it is one-to-one?
> 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
> a. Graph the function f(x) = sin(sin-1x) and explain the appearance of the graph. b. Graph the function g(x) = sin-1(sin x). How do you explain the appearance of this graph?
> Find the domain and range of the function g(x) = sin-1(3x + 1)
> Graph the given functions on the same screen. How are these graphs related? y = tan x, -π/ 2 < x < π/ 2; y = tan-1x; y = x
> 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
> Simplify the expression. sin(2arccosx)
> Simplify the expression. sin(tan-1x)
> Simplify the expression. tan(sin-1x)
> Prove that cos(sin-1 x) = 1 − x2 .
> Find the exact value of each expression. a. arcsin(sin(5π /4)) b. cos(2 sin-1 (5/13))
> Evaluate the difference quotient for the given function. Simplify your answer. f(a + h) – f(a) f(x) = x', h
> Find the exact value of each expression. a. cot-1(-√3 ) b. sec-1
> Find the exact value of each expression. a. sin-1(-1/√2 ) b. cos-1(√3/2)
> 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)
> When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electric charge given by Q(t) = Q0(1 – e-t/a) (The maximum charge capacity is Q0 and t is measured in seconds.) a. Find the inverse of this fun
> If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n = f(t) = 100 ∙ 2t/3. a. Find the inverse of this function and explain its meaning. b. When will the population reach 50,000
> a. If g(x) = x6 + x4, x ≥ 0, use a computer algebra system to find an expression for g-1(x). b. Use the expression in part (a) to graph y = t(x), y = x, and y = g-1(x) on the same screen.
> 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
> a. What are the values of eln 300 and ln(e300)? b. Use your calculator to evaluate eln 300 and ln(e300). What do you notice? Can you explain why the calculator has trouble?
> Evaluate the difference quotient for the given function. Simplify your answer. f(3 + h) – f(3) f(x) = 4 + 3x – x², h
> a. Find the domain of f(x) = ln(ex – 3). b. Find f -1 and its domain.
> Solve each inequality for x. a. 1 < e3x-1 < 2 b. 1 - 2 ln x < 3
> 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
> Solve each equation for x. a. 2x-5 = 3 b. ln x + ln(x – 1) − 1
> 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) = 4/3 πr3. 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
> Express the given quantity as a single logarithm. 1/3 ln(x + 2)3 + ½[ ln x – ln (x2 +3x + 2)2]
> Express the given quantity as a single logarithm. ln b + 2 ln c – 3 ln d
> Express the given quantity as a single logarithm. ln 10 + 2 ln 5
> Find the exact value of each expression. a. e-ln2 b. e ln(ln e 3)
> If f(x) = 3x2 - x + 2, find f(2), f(-2), f(a), f(-a), f(a + 1), 2f(a), f(2a), f(a2), [ f(a)]2, and f(a + h).
> Find the exact value of each expression. a. log1040 + log102.5 b. log860 - log83 - log85
> Find the exact value of each expression. a. log5 1/125 b. ln(1/e2)
> Find the exact value of each expression. a. log232 b. log82
> a. What is the natural logarithm? b. What is the common logarithm? c. Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.
> a. How is the logarithmic function y = logbx defined? b. What is the domain of this function? c. What is the range of this function? d. Sketch the general shape of the graph of the function y = logbx if b > 1.
