Chapter 3

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{(x+3)(x-2)}{x+1} \leq 0 $$

The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=\frac{1}{2}-2 x^{4} $$

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+1$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+1}{x+3}<2 $$

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-3(x-1)^{2}\left(x^{2}-4\right) $$

During the 1980 s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply- side economics, the theory uses functions such as $$ f(x)-\frac{80 x-8000}{x-110}, 30 \leq x \leq 100 $$ This function models the government tax revenue, \(f(x),\) in tens of billions of dollars, in terms of the tax rate, \(x\). The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases At a tax rate of (gasp) \(100 \%\), the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated. CAN'T COPY THE GRAPH a. Find and interpret \(f(30)\). Identify the solution as a point on the graph of the function. b. Rewrite the function by using long division to perform $$ (80 x-8000) \div(x-110) $$ Then use this new form of the function to find \(f(30) .\) Do you obtain the same answer as you did in part (a)? c. Is \(f\) a polynomial function? Explain your answer.

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x}{x-1}>2 $$

Galileo's telescope brought about revolutionary changes in astronomy. A comparable leap in our ability to observe the universe took place as a result of the Hubble Space Telescope. The space telescope was able to see stars and galaxies whose brightness is \(\frac{1}{s i}\) of the faintest objects observable using ground-based telescopes. Use the fact that the brightness of a point source, such as a star, varies inversely as the square of its distance from an observer to show that the space telescope was able to see about seven times farther than a ground-based telescope.

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-2(x-4)^{2}\left(x^{2}-25\right) $$

During the 1980 s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply- side economics, the theory uses functions such as $$ f(x)-\frac{80 x-8000}{x-110}, 30 \leq x \leq 100 $$ This function models the government tax revenue, \(f(x),\) in tens of billions of dollars, in terms of the tax rate, \(x\). The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases At a tax rate of (gasp) \(100 \%\), the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated. CAN'T COPY THE GRAPH a. Find and interpret \(f(30)\). Identify the solution as a point on the graph of the function. b. Rewrite the function by using long division to perform $$ (80 x-8000) \div(x-110) $$ Then use this new form of the function to find \(f(30) .\) Do you obtain the same answer as you did in part (a)? c. Is \(f\) a polynomial function? Explain your answer.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{2 x-1} \leq 3 $$

Begin by deciding on a product that interests the group because you are now in charge of advertising this product. Members were told that the demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. However, as more money is spent on advertising, the price of your product rises. Under what conditions would members recommend an increased expense in advertising? Once you've determined what your product is, write formulas for the given conditions and experiment with hypothetical numbers. What other factors might you take into consideration in terms of your recommendation? How do these factors affect the demand for your product?

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{2}(x-1)^{3}(x+2) $$

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

When the shot whose path is shown by the blue graph is released at an angle of \(35^{\circ},\) its height, \(f(x),\) in feet, can be modeled by $$ f(x)--0.01 x^{2}+0.7 x+6.1 $$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the blue graph. a. What is the maximum height of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from \(-3\) to 3 , inclusive, for \(x\). Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{1}{x-3}<1 $$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{3}(x+2)^{2}(x+1) $$

In your own words, state the Division Algorithm.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x-2}{x+2} \leq 2 $$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-x^{2}(x-1)(x+3) $$

How can the Division Algorithm be used to check the quotient and remainder in a long division problem?

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x)\), in feet, can be modeled by $$ f(x)--0.8 x^{2}+2.4 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x}{x+2} \geq 2 $$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=-x^{2}(x+2)(x-2) $$

Exercises \(53-60\) show incomplete graphs of given polynomial functions. a. Find all the zeros of each function. b. Without using a graphing utility, draw a complete graph of the function. $$ f(x)--5 x^{4}+4 x^{3}-19 x^{2}+16 x+4 $$ (GRAPH CANT COPY)

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x),\) in feet, can be modeled by $$ f(x)--0.8 x^{2}+3.2 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{2 x^{2}-5 x+2} $$

Among all pairs of numbers whose sum is \(16,\) find a pair whose product is as large as possible. What is the maximum product?

State the Remainder Theorem.

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\frac{1}{\sqrt{4 x^{2}-9 x+2}} $$

Among all pairs of numbers whose sum is \(20,\) find a pair whose product is as large as possible. What is the maximum product?

Explain how the Remainder Theorem can be used to find \(f(-6)\) if \(f(x)=x^{4}+7 x^{3}+8 x^{2}+11 x+5 .\) What advantage is there to using the Remainder Theorem in this situation rather than evaluating \(f(-6)\) directly?

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{\frac{2 x}{x+1}-1} $$

Among all pairs of numbers whose difference is \(16,\) find a pair whose product is as small as possible. What is the minimum product?

How can the Factor Theorem be used to determine if \(x-1\) is a factor of \(x^{3}-2 x^{2}-11 x+12 ?\)

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{\frac{x}{2 x-1}-1} $$

Among all pairs of numbers whose difference is 24 . find a pair whose product is as small as possible. What is the minimum product?

If you know that \(-2\) is a zero of $$ f(x)=x^{3}+7 x^{2}+4 x-12 $$ explain how to solve the equation $$ x^{3}+7 x^{2}+4 x-12=0 $$

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \left|x^{2}+2 x-36\right|>12 $$

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Writing in Mathematics Describe how to find the possible rational zeros of a polynomial function.

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \left|x^{2}+6 x+1\right|>8 $$

You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed? GRAPH CANNOT COPY.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division \(\left(x^{5}+1\right) \div(x+1),\) there's no need for me to follow all the steps involved in polynomial long division because 1 can work the problem in my head and sce that the quotient must be \(x^{4}+1\)

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{3}{x+3}>\frac{3}{x-2} $$

You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?