In Exercises 4750, do the following. (a) Determine the domains of and (b) Simplify and

Functions of the form where and are polynomials and is not the zero polynomial, are called ________ ________.

If as from the left or the right, then is a ________ ________ of the graph of

If as then is a ________ ________ of the graph of

For the rational function given by if the degree of is exactly one more than the degree of then the graph of has a ________ (or oblique) ________.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 512, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 1316, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 1316, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 1316, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 1316, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).]

In Exercises 1720, find the zeros (if any) of the rational function.

In Exercises 1720, find the zeros (if any) of the rational function.

In Exercises 1720, find the zeros (if any) of the rational function.

In Exercises 1720, find the zeros (if any) of the rational function. g_x_ _ x3 _ 8 x 2 _ 1

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2126, find the domain of the function and identify any horizontal and vertical asymptotes.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 2746, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 4750, do the following. (a) Determine the domains of and (b) Simplify and find any vertical asymptotes of the graph of (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and g.

In Exercises 4750, do the following. (a) Determine the domains of and (b) Simplify and find any vertical asymptotes of the graph of (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and g.

In Exercises 4750, do the following. (a) Determine the domains of and (b) Simplify and find any vertical asymptotes of the graph of (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and g.

In Exercises 4750, do the following. (a) Determine the domains of and (b) Simplify and find any vertical asymptotes of the graph of (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and g.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f _x_ _ 2x2 _ 5x _ 5 x _ 2

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 5164, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

In Exercises 6568, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line.

In Exercises 6568, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line.

In Exercises 6568, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line.

In Exercises 6568, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line.

In Exercises 6972, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6972, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6972, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6972, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

Pollution The cost (in millions of dollars) of removing of the industrial and municipal pollutants discharged into a river is given by (a) Use a graphing utility to graph the cost function. (b) Find the costs of removing 10%, 40%, and 75% of the pollutants. (c) According to this model, would it be possible to remove 100% of the pollutants? Explain.

Recycling In a pilot project, a rural township is given recycling bins for separating and storing recyclable products. The cost (in dollars) for supplying bins to of the population is given by (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to 15%, 50%, and 90% of the population. (c) According to this model, would it be possible to supply bins to 100% of the residents? Explain.

Population Growth The game commission introduces 100 deer into newly acquired state game lands. The population of the herd is modeled by where is the time in years (see figure). (a) Find the populations when and (b) What is the limiting size of the herd as time increases?

Concentration of a Mixture A 1000-liter tank contains 50 liters of a 25% brine solution. You add liters of a 75% brine solution to the tank. (a) Show that the concentration , the proportion of brine to total solution, in the final mixture is (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach?

Page Design A page that is inches wide and inches high contains 30 square inches of print. The top and bottom margins are 1 inch deep and the margins on each side are 2 inches wide (see figure). (a) Show that the total area on the page is (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility.

Page Design A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are inches wide. What should the dimensions of the page be so that the least amount of paper is used?

Average Speed A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 100 miles away. The average speeds for going and returning were and miles per hour, respectively. (a) Show that (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. (e) Are the results in the table what you expected? Explain. (f ) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

Sales The sales (in millions of dollars) for the Yankee Candle Company in the years 1998 through 2003 are shown in the table. (Source: The Yankee Candle Company) 1998 184.5 1999 256.6 2000 338.8 2001 379.8 2002 444.8 2003 508.6 A model for these data is given by where represents the year, with corresponding to 1998. (a) Use a graphing utility to plot the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to estimate the sales for the Yankee Candle Company in 2008. (c) Would this model be useful for estimating sales after 2008? Explain. True or False?

True or False? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. A polynomial can have infinitely many vertical asymptotes.

True or False? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes. Think About It

In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) Vertical asymptote: None Horizontal asymptote:

In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) Vertical asymptote: Horizontal asymptote: None

In Exercises 8588, completely factor the expression.

In Exercises 8588, completely factor the expression.

In Exercises 8588, completely factor the expression.

In Exercises 8588, completely factor the expression.

In Exercises 89-92, solve the inequality and graph thesolution on the real number line.

In Exercises 89-92, solve the inequality and graph thesolution on the real number line.

In Exercises 89-92, solve the inequality and graph thesolution on the real number line.

In Exercises 89-92, solve the inequality and graph thesolution on the real number line.

Make a Decision To work an extended application analyzing the total manpower of the Department of Defense, visit this texts website at college.hmco.com. (Data Source: U.S. Census Bureau)