Let and be linear functions with equations f (x) = mx + b1 and g (x) m2x + b2. Is f 0 g also a linear function? If so, what is the slope of its graph?
> The graph of f is given. Use it to graph the following functions. (a). y = f (2x) (b). y = f (1/2x) (c). y = f (-x) (d). y = -f (-x) 1. 1
> Find an expression for the function whose graph is the given curve. The bottom half of the parabola x + (y -1)2 = 0
> The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; is measured in hours starting at midnight.) (a). What was the power consumption at 6 AM? At 6 PM? (b). When was the power consumption the lowest
> The city of New Orleans is located at latitude 300N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun
> The table gives the winning heights for the Olympic pole vault competitions up to the year 2000. (a). Make a scatter plot and decide whether a linear model is appropriate. (b). Find and graph the regression line. (c). Use the linear model to predict th
> Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. (a). Make a scatter plot of the data. (b). Find and graph the regression
> Find the domain and sketch the graph of the function. f (x) = {(3 -1/2x & if & x > 0 @ 2x -5 & if & x > 0)
> The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey. (a). Make a scatter plot of these data and decide whether a linear model is appropriate. (b). Find and gr
> Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at and 173 0F chirps per minute at 800F. (a). Find
> The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = 9/5C +32. (a). Sketch a graph of this function. (b). What is the slope of the graph and what does it represent? What is the F-intercept and
> Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration functio
> The graph of is given. Draw the graphs of the following functions. (a). y = f (x) -2 (b). y = f (x -2) (c). y = -2 f(x) (d). y = f (1/3x) + 1 2 1
> Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race? у (m) A B C 100 20 t
> The first graph in the figure is that of y = sin 45 x as displayed by a TI-83 graphing calculator. It is inaccurate and so, to help explain its appearance, we replot the curve in dot mode in the second graph. What two sine curves does the calculator appe
> The figure shows the graphs of y = sin 96 x and y = sin 2x as displayed by a TI-83 graphing calculator. The first graph is inaccurate. Explain why the two graphs appear identical. [Hint: The TI-83’s graphing window is 95 pixels wide. Wh
> Find expressions for the quadratic functions whose graphs are shown. (-2, 2), (0, 1) (4, 2) 3 4(1, –2.5)
> What happens to the graph of the equation y2 = cx3 + x2 as c varies?
> (a). Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f (2) = 1 and sketch several members of the family. (c). Which function belo
> Find the domain of the function. h (x) = 1/4√x2 - 5x
> Graph the function f (x) = √1 + cx2 for various values of c. Describe how changing the value of c affects the graph.
> Graph the function f (x) = x4 + cx2 + x for several values of c. How does the graph change when changes?
> In this exercise we consider the family of functions f (x) = 1/xn, where is a positive integer. (a). Graph the functions y = 1/x and y = 1/x3 on the same screen using the viewing rectangle [-3, 3] by [-3, 3]. (b). Graph the functions y = 1/x2 and y = 1/x
> The graph of is given. Match each equation with its graph and give reasons for your choices. (a). y = f (x – 4) (b). y = f (x) – 3 (c). y = 1/3 f (x) (d). y = -f (x + 4) (e). y = 2f (x + 6) y. 6 3 -6 -3 3. -3
> In this exercise we consider the family of root functions f (x) = n√x, where is a positive integer. (a). Graph the functions y = √x, y = 4√x and y = 6√x on the same screen using the viewing rectangle [-1, 4] by [-1, 3]. (b). Graph the functions y = x, y
> Use the given graph of f to sketch the graph of y =1/f (x). Which features of are the most important in sketching y = 1/f (x)? Explain how they are used. 1
> For what values of is it true that |sin x – x| < 0.1?
> A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its b
> Use graphs to determine which of the functions f (x) = 10x2 and g (x) = x3/10 is eventually larger (that is, larger when is very large).
> The table shows the percentage of the population of Argentina that has lived in rural areas from 1955 to 2000. Find a model for the data and use it to estimate the rural percentage in 1988 and 2002. Percentage Percentage Year rural Year rural 1955 3
> Find all solutions of the equation correct to two decimal places. tan x = √1 - x2
> Find all solutions of the equation correct to two decimal places. √x = x3 - 1
> Find all solutions of the equation correct to two decimal places. x4 – x = 1
> Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
> Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
> Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x) = 1 + 3x? – x*
> For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. (a) (b) y. y
> Explain how each graph is obtained from the graph of y = f (x). (a). y = f (x) + 8 (b). y =y f (x + 8) (c). y = 8 f (x) (d). y = f (8x) (e). y = f (-x) -1 (f). y = 8f(1/8x)
> Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x) = x|x|
> Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x = х+1
> Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x) x* + 1
> Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x) = x? + 1 -2
> Sketch a rough graph of the number of hours of daylight as a function of the time of year.
> Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 1
> You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the ela
> Suppose g is an even function and let h = f 0 g. Is h always an even function?
> If f (x) = x + 4 and h (x) = 4x - 1, find a function g such that h.
> If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c = 0.041D (a + 1). Suppose the dosage for an adult is 200 mg. (a). Find the slope of the graph of c. W
> For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. (а) (b) y y
> If you invest dollars at 4% interest compounded annually, then the amount A (x) of the investment after one year is What do these compositions represent? Find a formula for the composition of n copies of A. A(x) = 1.04x. Find A • A, A • A • A, and
> The Heaviside function defined in Exercise 57 can also be used to define the ramp function y = ct H (t), which represents a gradual increase in voltage or current in a circuit. Exercise 57: The Heaviside function H is defined by It is used in the stu
> Find a formula for the described function and state its domain. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.
> An airplane is flying at a speed of at an altitude of 350 mi/h one mile and passes directly over a radar station at time t = 0. (a). Express the horizontal distance (in miles) that the plane has flown as a function of t. (b). Express the distance between
> Find a formula for the described function and state its domain. Express the area of an equilateral triangle as a function of the length of a side.
> A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s. (a). Express the radius of the balloon as a function of the time t (in seconds). (b). If V is the volume of the balloon as a function of the radius, fi
> A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 m/s. (a). Express the radius of this circle as a function of the time (in seconds). (b). If A is the area of this circle as a function of the radius, find A
> Use the given graphs of f and g to estimate the value of f (g (x)) for x = -5, -4, -3…5. Use these estimates to sketch a rough graph of f 0 g. 지
> Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.002t + 8.50, where T is temperature in 0Cand represents years since 1900. (a). W
> The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a). Express the monthly cost C as a function of the distance driven d assuming
> Use the table to evaluate each expression. 3 4 1 5 6. f(x) 3 4 5 9(x) 6 1 3. 2. 2. 2. 2. 2. 3. (a) f(g(1)) (d) g(g(1)) (b) g(f(1)) (e) (gof)(3) (c) f(f(1)) (f) (fo g)(6)
> Express the function in the form f o g 0 h. H (x) = sec4 (√x)
> Express the function in the form f o g 0 h. H (x) = 8√2 + |x|
> Express the function in the form f o g 0 h. H (x) = 1 - 3x2
> Express the function in the form fog. u (t) = tan 6/1 + tan t
> Express the function in the form f0g. u (t) = √cos t
> Express the function in the form f0g. G (x) = 3√x/1+x
> Express the function in the form f0g. F (x) = 3√x/1 + 3√x
> Express the function in the form f0g. F (x) = cos2x
> Express the function in the form f0g. F (x) = (2x + x2)4
> At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface, the water pressure increases by 4.34 lb/in2 for every 10 ft of descent. (a). Express the water pressure as a function of the de
> Find f0g0h. f (x) = tan x, g (x) = x/x-1, h (x) = 3√x
> Find f0g0h. f (x) = √x - 3 g (x) = x2 h (x) = x3 + 2
> Find f0g0h. f (x) = 2x - 1 g (x) = x2 h (x) = 1 - x
> Find f0g0h. f (x) = x + 1 g(x) = 2x h (x) = x - 1
> This exercise explores the effect of the inner function on a composite function y = f (g (x)). (a). Graph the function y = sin (√x) using the viewing rectangle [0, 400] by [-1.5, 1.5]. How does this graph differ from the graph of the sine function? (b).
> Find the domain and sketch the graph of the function. f (x) = 2 - 0.4x
> The curves with equations y = |x| / √c – x2 are called bullet-nose curves. Graph some of these curves to see why. What happens as c increases?
> Graph the function y = xn2-x, x > 0, for n = 1, 2 ,3, 4, 5 and 6. How does the graph change as n increases?
> Find the functions (a) f0g, (b) g0f, (c) f0f, and (d)g0g and their domains. f(x) — х — 2, д() — х* + Зх + 4
> Find the functions (a) f0g, (b) g0f, (c) f0f, and (d)g0g and their domains. f(x) = x ? - 1, g(x) = 2x + 1
> The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a). Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch th
> Find (a) f + g, (b) f - g, (c) fg, and (d) f/g and state their domains. f(x) = /3 – x, g(x) = /x² – 1
> Find (a) f + g, (b) f - g, (c) fg, and (d) f/g and state their domains. f(x) = x³ + 2x?, g(x) = 3x² – 1
> Graph the polynomials P (x) = 3x2 – 5x3 + 2x and Q (x) = 3x5 on the same screen, first using the viewing rectangle [-2, 2] by [-2, 2] and then changing to [-10, 10] by [-10000, 10000]. What do you observe from these graphs?
> (a). How is the graph of y = f (|x|) related to the graph of f? (b). Sketch the graph of y = sin |x|. (c). Sketch the graph of y = √|x|
> Use graphs to determine which of the functions f (x) = x4 - 100x3 and g (x) = x3 is eventually larger.
> Use the data in the table to model the population of the world in the 20th century by a cubic function. Then use your model to estimate the population in the year 1925. Population (millions) Population (millions) Year Year 1900 1650 1960 3040 1910 1
> We saw in Example 9 that the equation cos x = x has exactly one solution. (a) Use a graph to show that the equation cos x = 0.3 x has three solutions and find their values correct to two decimal places. (b) Find an approximate value of m such that the eq
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. tan 4 x 4
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = |x – 2|
> (a). Try to find an appropriate viewing rectangle for f (x) = (x – 10)3 2-x. (b). Do you need more than one window? Why?
> Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? у — 6 — 4х — х, у — Зr + 18; [-6, 2] by [-5, 20]
> Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? у — Зx? — бх + 1, у-0.23х — 2.25; [-1, 3] by [-2.5, 1.5]
> Graph the hyperbola by graphing the functions y2 – 9x2 = 1 whose graphs are the upper and lower branches of the hyperbola.
> Graph the ellipse by graphing the functions 4x2 + 2y2 = 1 whose graphs are the upper and lower halves of the ellipse.
> Graph the function f (x) = x2 √30 – x in an appropriate viewing rectangle. Why does part of the graph appear to be missing?
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = sin (x/2)
> Determine an appropriate viewing rectangle for the given function and use it to draw the graph. y = x2 + 0.02 sin 50x
> Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = 1 + 2 cos x
> Determine an appropriate viewing rectangle for the given function and use it to draw the graph. f(x) = sec(207x)
