91Ó°ÊÓ

Functions of the form \(f(x)=N(x) / D(x),\) where \(N(x)\) and \(D(x)\) are polynomials and \(D(x)\) is not the zero polynomial, are called _________ _________.

Finding the Domain of a Rational Function In Exercises \(5-8,\) find the domain of the function and discuss the behavior of \(f\) near any excluded \(x\) -values. $$f(x)=\frac{1}{x-1}$$

Finding Vertical and Horizontal Asymptotes In Exercises \(9-16,\) find all vertical and horizontal asymptotes of the graph of the function. $$f(x)=\frac{3 x^{2}+1}{x^{2}+x+9}$$

Sketching Graphs of Quadratic Functions In }} \\ {\text { Exercises } 13-16, \text { sketch the graph of each quadratic }} \\ {\text { function and compare it with the graph of } y=x^{2} .}\end{array} $$ \begin{array}{ll}{\text { (a) } f(x)=(x-1)^{2}} & {\text { (b) } g(x)=(3 x)^{2}+1} \\ {\text { (c) } h(x)=\left(\frac{1}{3} x\right)^{2}-3} & {\text { (d) } k(x)=(x+3)^{2}}\end{array} $$

Compound Interest In Exercises 15 and \(16,\) determine the time necessary for \(P\) dollars to double when it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$r=10 \%$$

Website Growth The number \(y\) of hits a new website receives each month can be modeled by \(y=4080 e^{k t},\) where \(t\) represents the number of months the website has been operating. In the website's thirdmonth, there were \(10,000\) hits. Find the value of \(k,\) and use this value to predict the number of hits the website will receive after 24 months.

Radioactive Decay Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \(^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of 14 C absorbed by a tree growing today. A piece of ancient charcoal contains only 15\(\%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal, assuming that the half-life of \(^{14} \mathrm{C}\) is 5715 years?

Solving a Rational Inequality In Exercises \(39-52\) , solve the inequality. Then graph the solution set. $$\frac{3 x-5}{x-5} \geq 0$$

Population Growth A conservation organization released 100 animals of an endangered species into a game preserve. The preserve has a carrying capacity of 1000 animals. The growth of the pack is modeled by the logistic curve $$p(t)=\frac{1000}{1+9 e^{-0.1656 t}}$$ where \(t\) is measured in months (see figure). (a) Estimate the population after 5 months. (b) After how many months is the population 500\(?\) (c) Use a graphing utility to graph the function. Use thh graph to determine the horizontal asymptotes, and interpret the meaning of the asymptotes in the context of the problem.

Intensity of Sound In Exercises \(47-50\) , use the following information for determining sound intensity. The level of sound \(\beta,\) in decibels, with an intensity of \(I,\) is given by \(\beta=10 \log \left(I I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 47 and \(48,\) find the level of sound \(\beta .\) (a) \(I=10^{-10}\) watt per \(\mathrm{m}^{2}\) (quiet room) (b) \(I=10^{-5}\) watt per \(\mathrm{m}^{2}\) (busy street corner) (c) \(I=10^{-8}\) watt per \(\mathrm{m}^{2}\) (quiet radio) (d) \(I=10^{0}\) watt per \(\mathrm{m}^{2}\) (threshold of pain)